# Properties

 Label 32-300e16-1.1-c10e16-0-0 Degree $32$ Conductor $4.305\times 10^{39}$ Sign $1$ Analytic cond. $3.03548\times 10^{36}$ Root an. cond. $13.8060$ Motivic weight $10$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 − 3.31e5·11-s − 1.40e8·31-s + 2.95e7·41-s + 3.57e9·61-s + 1.85e9·71-s − 1.54e9·81-s − 1.89e10·101-s − 1.48e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
 L(s)  = 1 − 2.05·11-s − 4.91·31-s + 0.255·41-s + 4.23·61-s + 1.02·71-s − 4/9·81-s − 1.80·101-s − 5.73·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{32} \cdot 3^{16} \cdot 5^{32}$$ Sign: $1$ Analytic conductor: $$3.03548\times 10^{36}$$ Root analytic conductor: $$13.8060$$ Motivic weight: $$10$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{300} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [5]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.0003519601633$$ $$L(\frac12)$$ $$\approx$$ $$0.0003519601633$$ $$L(6)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$( 1 + p^{18} T^{4} )^{4}$$
5 $$1$$
good7 $$1 - 197292832374494404 T^{4} +$$$$16\!\cdots\!10$$$$T^{8} -$$$$80\!\cdots\!16$$$$p^{4} T^{12} +$$$$37\!\cdots\!19$$$$p^{8} T^{16} -$$$$80\!\cdots\!16$$$$p^{44} T^{20} +$$$$16\!\cdots\!10$$$$p^{80} T^{24} - 197292832374494404 p^{120} T^{28} + p^{160} T^{32}$$
11 $$( 1 + 82776 T + 54314315020 T^{2} + 3977091470105064 T^{3} +$$$$14\!\cdots\!34$$$$T^{4} + 3977091470105064 p^{10} T^{5} + 54314315020 p^{20} T^{6} + 82776 p^{30} T^{7} + p^{40} T^{8} )^{4}$$
13 $$1 +$$$$19\!\cdots\!08$$$$T^{4} +$$$$29\!\cdots\!78$$$$T^{8} +$$$$28\!\cdots\!56$$$$T^{12} -$$$$49\!\cdots\!05$$$$T^{16} +$$$$28\!\cdots\!56$$$$p^{40} T^{20} +$$$$29\!\cdots\!78$$$$p^{80} T^{24} +$$$$19\!\cdots\!08$$$$p^{120} T^{28} + p^{160} T^{32}$$
17 $$1 -$$$$18\!\cdots\!64$$$$T^{4} +$$$$17\!\cdots\!40$$$$T^{8} -$$$$11\!\cdots\!76$$$$T^{12} +$$$$53\!\cdots\!14$$$$T^{16} -$$$$11\!\cdots\!76$$$$p^{40} T^{20} +$$$$17\!\cdots\!40$$$$p^{80} T^{24} -$$$$18\!\cdots\!64$$$$p^{120} T^{28} + p^{160} T^{32}$$
19 $$( 1 - 12159415166308 T^{2} +$$$$13\!\cdots\!78$$$$T^{4} -$$$$86\!\cdots\!56$$$$T^{6} +$$$$63\!\cdots\!95$$$$T^{8} -$$$$86\!\cdots\!56$$$$p^{20} T^{10} +$$$$13\!\cdots\!78$$$$p^{40} T^{12} - 12159415166308 p^{60} T^{14} + p^{80} T^{16} )^{2}$$
23 $$1 -$$$$46\!\cdots\!08$$$$T^{4} +$$$$11\!\cdots\!28$$$$T^{8} -$$$$21\!\cdots\!56$$$$T^{12} +$$$$35\!\cdots\!70$$$$T^{16} -$$$$21\!\cdots\!56$$$$p^{40} T^{20} +$$$$11\!\cdots\!28$$$$p^{80} T^{24} -$$$$46\!\cdots\!08$$$$p^{120} T^{28} + p^{160} T^{32}$$
29 $$( 1 - 2702520384392792 T^{2} +$$$$33\!\cdots\!28$$$$T^{4} -$$$$24\!\cdots\!44$$$$T^{6} +$$$$12\!\cdots\!70$$$$T^{8} -$$$$24\!\cdots\!44$$$$p^{20} T^{10} +$$$$33\!\cdots\!28$$$$p^{40} T^{12} - 2702520384392792 p^{60} T^{14} + p^{80} T^{16} )^{2}$$
31 $$( 1 + 35201204 T + 1416319277340010 T^{2} +$$$$24\!\cdots\!16$$$$T^{3} +$$$$54\!\cdots\!19$$$$T^{4} +$$$$24\!\cdots\!16$$$$p^{10} T^{5} + 1416319277340010 p^{20} T^{6} + 35201204 p^{30} T^{7} + p^{40} T^{8} )^{4}$$
37 $$1 -$$$$10\!\cdots\!08$$$$T^{4} +$$$$53\!\cdots\!28$$$$T^{8} -$$$$19\!\cdots\!56$$$$T^{12} +$$$$50\!\cdots\!70$$$$T^{16} -$$$$19\!\cdots\!56$$$$p^{40} T^{20} +$$$$53\!\cdots\!28$$$$p^{80} T^{24} -$$$$10\!\cdots\!08$$$$p^{120} T^{28} + p^{160} T^{32}$$
41 $$( 1 - 7388400 T + 20886066228918004 T^{2} -$$$$16\!\cdots\!00$$$$T^{3} +$$$$28\!\cdots\!06$$$$T^{4} -$$$$16\!\cdots\!00$$$$p^{10} T^{5} + 20886066228918004 p^{20} T^{6} - 7388400 p^{30} T^{7} + p^{40} T^{8} )^{4}$$
43 $$1 -$$$$84\!\cdots\!08$$$$T^{4} +$$$$75\!\cdots\!78$$$$T^{8} -$$$$39\!\cdots\!56$$$$T^{12} +$$$$23\!\cdots\!95$$$$T^{16} -$$$$39\!\cdots\!56$$$$p^{40} T^{20} +$$$$75\!\cdots\!78$$$$p^{80} T^{24} -$$$$84\!\cdots\!08$$$$p^{120} T^{28} + p^{160} T^{32}$$
47 $$1 -$$$$14\!\cdots\!44$$$$T^{4} +$$$$11\!\cdots\!80$$$$T^{8} -$$$$53\!\cdots\!56$$$$T^{12} +$$$$17\!\cdots\!74$$$$T^{16} -$$$$53\!\cdots\!56$$$$p^{40} T^{20} +$$$$11\!\cdots\!80$$$$p^{80} T^{24} -$$$$14\!\cdots\!44$$$$p^{120} T^{28} + p^{160} T^{32}$$
53 $$1 +$$$$35\!\cdots\!56$$$$T^{4} +$$$$10\!\cdots\!80$$$$T^{8} -$$$$28\!\cdots\!76$$$$p^{4} T^{12} -$$$$15\!\cdots\!66$$$$p^{8} T^{16} -$$$$28\!\cdots\!76$$$$p^{44} T^{20} +$$$$10\!\cdots\!80$$$$p^{80} T^{24} +$$$$35\!\cdots\!56$$$$p^{120} T^{28} + p^{160} T^{32}$$
59 $$( 1 - 2948603619524520632 T^{2} +$$$$39\!\cdots\!88$$$$T^{4} -$$$$33\!\cdots\!44$$$$T^{6} +$$$$19\!\cdots\!70$$$$T^{8} -$$$$33\!\cdots\!44$$$$p^{20} T^{10} +$$$$39\!\cdots\!88$$$$p^{40} T^{12} - 2948603619524520632 p^{60} T^{14} + p^{80} T^{16} )^{2}$$
61 $$( 1 - 894215708 T + 1013888684766468778 T^{2} -$$$$91\!\cdots\!56$$$$T^{3} +$$$$71\!\cdots\!95$$$$T^{4} -$$$$91\!\cdots\!56$$$$p^{10} T^{5} + 1013888684766468778 p^{20} T^{6} - 894215708 p^{30} T^{7} + p^{40} T^{8} )^{4}$$
67 $$1 +$$$$12\!\cdots\!76$$$$T^{4} +$$$$71\!\cdots\!70$$$$T^{8} +$$$$22\!\cdots\!64$$$$T^{12} +$$$$64\!\cdots\!59$$$$T^{16} +$$$$22\!\cdots\!64$$$$p^{40} T^{20} +$$$$71\!\cdots\!70$$$$p^{80} T^{24} +$$$$12\!\cdots\!76$$$$p^{120} T^{28} + p^{160} T^{32}$$
71 $$( 1 - 463298160 T + 3205719637261167604 T^{2} +$$$$45\!\cdots\!20$$$$T^{3} -$$$$21\!\cdots\!94$$$$T^{4} +$$$$45\!\cdots\!20$$$$p^{10} T^{5} + 3205719637261167604 p^{20} T^{6} - 463298160 p^{30} T^{7} + p^{40} T^{8} )^{4}$$
73 $$1 +$$$$34\!\cdots\!36$$$$T^{4} +$$$$41\!\cdots\!40$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{12} -$$$$43\!\cdots\!86$$$$T^{16} -$$$$13\!\cdots\!76$$$$p^{40} T^{20} +$$$$41\!\cdots\!40$$$$p^{80} T^{24} +$$$$34\!\cdots\!36$$$$p^{120} T^{28} + p^{160} T^{32}$$
79 $$( 1 - 52587262505497101304 T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$18\!\cdots\!16$$$$T^{6} +$$$$20\!\cdots\!94$$$$T^{8} -$$$$18\!\cdots\!16$$$$p^{20} T^{10} +$$$$12\!\cdots\!60$$$$p^{40} T^{12} - 52587262505497101304 p^{60} T^{14} + p^{80} T^{16} )^{2}$$
83 $$1 +$$$$13\!\cdots\!92$$$$T^{4} +$$$$15\!\cdots\!28$$$$T^{8} +$$$$25\!\cdots\!44$$$$T^{12} +$$$$11\!\cdots\!70$$$$T^{16} +$$$$25\!\cdots\!44$$$$p^{40} T^{20} +$$$$15\!\cdots\!28$$$$p^{80} T^{24} +$$$$13\!\cdots\!92$$$$p^{120} T^{28} + p^{160} T^{32}$$
89 $$( 1 - 97327905405050174408 T^{2} +$$$$62\!\cdots\!28$$$$T^{4} -$$$$26\!\cdots\!56$$$$T^{6} +$$$$92\!\cdots\!70$$$$T^{8} -$$$$26\!\cdots\!56$$$$p^{20} T^{10} +$$$$62\!\cdots\!28$$$$p^{40} T^{12} - 97327905405050174408 p^{60} T^{14} + p^{80} T^{16} )^{2}$$
97 $$1 +$$$$18\!\cdots\!92$$$$T^{4} +$$$$17\!\cdots\!78$$$$T^{8} +$$$$14\!\cdots\!44$$$$T^{12} +$$$$12\!\cdots\!95$$$$T^{16} +$$$$14\!\cdots\!44$$$$p^{40} T^{20} +$$$$17\!\cdots\!78$$$$p^{80} T^{24} +$$$$18\!\cdots\!92$$$$p^{120} T^{28} + p^{160} T^{32}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−1.77399157050840749169448939855, −1.68870756677645734754313475189, −1.61846909695636509824643086209, −1.61306694460312354859928519503, −1.50384031707441918479476374739, −1.40210376971215513004586222796, −1.34454216797751286127406579960, −1.24444609141497052646315171457, −1.24173785114494554175147296270, −1.17651289213584859133700699904, −1.05252320846996518449768010455, −0.977023754214709853436064580166, −0.970191627561600650688329368094, −0.958677837887395316976400200072, −0.903405264452300844188332496236, −0.76905404561731283638452951763, −0.71691297522869950941342765177, −0.44605562088508565210871805378, −0.38831492963380222619969679053, −0.32010713503366265159692249853, −0.19622741202829308383502579333, −0.11291979207633946709021935201, −0.05761272290755584563131266527, −0.04735510250533229522833412561, −0.02060980301419660918878602185, 0.02060980301419660918878602185, 0.04735510250533229522833412561, 0.05761272290755584563131266527, 0.11291979207633946709021935201, 0.19622741202829308383502579333, 0.32010713503366265159692249853, 0.38831492963380222619969679053, 0.44605562088508565210871805378, 0.71691297522869950941342765177, 0.76905404561731283638452951763, 0.903405264452300844188332496236, 0.958677837887395316976400200072, 0.970191627561600650688329368094, 0.977023754214709853436064580166, 1.05252320846996518449768010455, 1.17651289213584859133700699904, 1.24173785114494554175147296270, 1.24444609141497052646315171457, 1.34454216797751286127406579960, 1.40210376971215513004586222796, 1.50384031707441918479476374739, 1.61306694460312354859928519503, 1.61846909695636509824643086209, 1.68870756677645734754313475189, 1.77399157050840749169448939855

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.