Properties

Label 4-2850e2-1.1-c1e2-0-12
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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 − 9-s + 16-s − 2·19-s + 12·29-s + 4·31-s + 36-s + 10·49-s + 12·59-s + 4·61-s − 64-s + 2·76-s − 4·79-s + 81-s + 24·89-s − 12·101-s + 32·109-s − 12·116-s − 22·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.458·19-s + 2.22·29-s + 0.718·31-s + 1/6·36-s + 10/7·49-s + 1.56·59-s + 0.512·61-s − 1/8·64-s + 0.229·76-s − 0.450·79-s + 1/9·81-s + 2.54·89-s − 1.19·101-s + 3.06·109-s − 1.11·116-s − 2·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.198803122\)
\(L(\frac12)\) \(\approx\) \(2.198803122\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.929250555233847252998844921940, −8.561509175501574114938672167567, −8.285040979478221438701947071185, −8.006389099924641886637409563523, −7.48237136900885735434735428006, −7.03622826241972808423931987005, −6.73895089475014439347574311960, −6.17965903299152382001666092149, −6.06754921632017686259687276858, −5.43165905903612833945625216620, −5.08933773316560704934322091353, −4.62519732755036486478567464564, −4.37663369024782113596086248540, −3.75474903602742596834981645967, −3.47751733069437604939025681896, −2.66165415259016281952066309499, −2.58838930687612867925045891981, −1.84792951982160529378242756395, −1.00505238716510504606513496644, −0.58327489241786780773589090378, 0.58327489241786780773589090378, 1.00505238716510504606513496644, 1.84792951982160529378242756395, 2.58838930687612867925045891981, 2.66165415259016281952066309499, 3.47751733069437604939025681896, 3.75474903602742596834981645967, 4.37663369024782113596086248540, 4.62519732755036486478567464564, 5.08933773316560704934322091353, 5.43165905903612833945625216620, 6.06754921632017686259687276858, 6.17965903299152382001666092149, 6.73895089475014439347574311960, 7.03622826241972808423931987005, 7.48237136900885735434735428006, 8.006389099924641886637409563523, 8.285040979478221438701947071185, 8.561509175501574114938672167567, 8.929250555233847252998844921940

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