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Properties

Label 4-990e2-1.1-c1e2-0-12
Degree $4$
Conductor $980100$
Sign $1$
Analytic cond. $62.4920$
Root an. cond. $2.81161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s + 2·11-s + 16-s + 14·19-s + 4·20-s + 11·25-s + 6·29-s − 10·31-s + 16·41-s − 2·44-s + 13·49-s − 8·55-s + 24·59-s + 10·61-s − 64-s + 14·71-s − 14·76-s + 8·79-s − 4·80-s − 14·89-s − 56·95-s − 11·100-s − 12·101-s − 20·109-s − 6·116-s + 3·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s + 0.603·11-s + 1/4·16-s + 3.21·19-s + 0.894·20-s + 11/5·25-s + 1.11·29-s − 1.79·31-s + 2.49·41-s − 0.301·44-s + 13/7·49-s − 1.07·55-s + 3.12·59-s + 1.28·61-s − 1/8·64-s + 1.66·71-s − 1.60·76-s + 0.900·79-s − 0.447·80-s − 1.48·89-s − 5.74·95-s − 1.09·100-s − 1.19·101-s − 1.91·109-s − 0.557·116-s + 3/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(980100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(62.4920\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 980100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589224869\)
\(L(\frac12)\) \(\approx\) \(1.589224869\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.17.a_aj
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.19.ao_dj
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.37.a_acv
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \) 2.53.a_aeb
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.61.ak_fr
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.89.o_it
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.11975913934081495270256972200, −9.660639654565721256077470980068, −9.329256672519071250659282695731, −9.024055290109692232246052116111, −8.347412845105778811456386445898, −8.252739761647479206996688474912, −7.51005772481090775029855781177, −7.41184876367244555114007969484, −7.04997524955835712319216218633, −6.55305862079604024993534217601, −5.66507215344492687806563123059, −5.33784860937264823323134051463, −5.09483809688426717572736246329, −4.15468020571758649400512104711, −4.01403137655790120099080331250, −3.60022368666345842948047263111, −3.02245347540903862831709389818, −2.41984865617263889791798512678, −1.02061570736468428199771033936, −0.803406419070543251268829435479, 0.803406419070543251268829435479, 1.02061570736468428199771033936, 2.41984865617263889791798512678, 3.02245347540903862831709389818, 3.60022368666345842948047263111, 4.01403137655790120099080331250, 4.15468020571758649400512104711, 5.09483809688426717572736246329, 5.33784860937264823323134051463, 5.66507215344492687806563123059, 6.55305862079604024993534217601, 7.04997524955835712319216218633, 7.41184876367244555114007969484, 7.51005772481090775029855781177, 8.252739761647479206996688474912, 8.347412845105778811456386445898, 9.024055290109692232246052116111, 9.329256672519071250659282695731, 9.660639654565721256077470980068, 10.11975913934081495270256972200

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