Properties

Degree $4$
Conductor $28900$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 5·9-s − 12·11-s + 16-s + 14·19-s − 2·20-s − 25-s + 10·29-s + 10·31-s − 5·36-s + 12·44-s + 10·45-s + 14·49-s − 24·55-s − 10·59-s − 6·61-s − 64-s − 30·71-s − 14·76-s − 16·79-s + 2·80-s + 16·81-s + 2·89-s + 28·95-s − 60·99-s + 100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 5/3·9-s − 3.61·11-s + 1/4·16-s + 3.21·19-s − 0.447·20-s − 1/5·25-s + 1.85·29-s + 1.79·31-s − 5/6·36-s + 1.80·44-s + 1.49·45-s + 2·49-s − 3.23·55-s − 1.30·59-s − 0.768·61-s − 1/8·64-s − 3.56·71-s − 1.60·76-s − 1.80·79-s + 0.223·80-s + 16/9·81-s + 0.211·89-s + 2.87·95-s − 6.03·99-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{170} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28877\)
\(L(\frac12)\) \(\approx\) \(1.28877\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
17$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
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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

−13.45019682525548528185759574713, −12.56902754620698119956140156272, −12.19723532688957565209501873946, −11.68846318239185755529641597631, −10.65836645624033867377691971627, −10.24808373329158249543853928376, −10.18243183921051514266936658515, −9.756618677118959182678570781437, −9.132870746615799306542864492406, −8.264859617764210124365604697928, −7.70620778082021457200679001867, −7.54852750955059439015496471914, −6.88711588675284328701798294168, −5.69149124039611196796630011913, −5.58041120481242809972795257783, −4.81233862370992299921138409531, −4.48216253547213087813980636822, −2.91003541838342100486717755651, −2.78849164707539717056843579898, −1.24620747171930182430763415435, 1.24620747171930182430763415435, 2.78849164707539717056843579898, 2.91003541838342100486717755651, 4.48216253547213087813980636822, 4.81233862370992299921138409531, 5.58041120481242809972795257783, 5.69149124039611196796630011913, 6.88711588675284328701798294168, 7.54852750955059439015496471914, 7.70620778082021457200679001867, 8.264859617764210124365604697928, 9.132870746615799306542864492406, 9.756618677118959182678570781437, 10.18243183921051514266936658515, 10.24808373329158249543853928376, 10.65836645624033867377691971627, 11.68846318239185755529641597631, 12.19723532688957565209501873946, 12.56902754620698119956140156272, 13.45019682525548528185759574713

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