Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2} $
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 − 4·11-s + 16-s − 8·19-s + 2·29-s + 6·31-s + 6·41-s + 4·44-s − 49-s − 6·59-s + 10·61-s − 64-s − 8·71-s + 8·76-s + 4·79-s + 20·89-s + 8·109-s − 2·116-s − 10·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.20·11-s + 1/4·16-s − 1.83·19-s + 0.371·29-s + 1.07·31-s + 0.937·41-s + 0.603·44-s − 1/7·49-s − 0.781·59-s + 1.28·61-s − 1/8·64-s − 0.949·71-s + 0.917·76-s + 0.450·79-s + 2.11·89-s + 0.766·109-s − 0.185·116-s − 0.909·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9922500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.206515804$
$L(\frac12)$  $\approx$  $1.206515804$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.902186558754098285799163241864, −8.441209391517775762363458788091, −8.056352970138622546857262447687, −7.917622718233711942763675610896, −7.52875966659886035518224209237, −6.93824515043766526694219211083, −6.51388161801017148728456699899, −6.36490624378062815697104960504, −5.62609604427547737567948586981, −5.62442206046540222346720984564, −4.85295102670311579135486305208, −4.71876505107304421530976423227, −4.16767890302369733025766307495, −3.96838017352575114936592203683, −3.08103096442649605749197867140, −2.98695522803573316007910573951, −2.17203488202553299466463835934, −2.04139695684873212623534625709, −1.03404631717384617889362410988, −0.38891980352997469944515553764, 0.38891980352997469944515553764, 1.03404631717384617889362410988, 2.04139695684873212623534625709, 2.17203488202553299466463835934, 2.98695522803573316007910573951, 3.08103096442649605749197867140, 3.96838017352575114936592203683, 4.16767890302369733025766307495, 4.71876505107304421530976423227, 4.85295102670311579135486305208, 5.62442206046540222346720984564, 5.62609604427547737567948586981, 6.36490624378062815697104960504, 6.51388161801017148728456699899, 6.93824515043766526694219211083, 7.52875966659886035518224209237, 7.917622718233711942763675610896, 8.056352970138622546857262447687, 8.441209391517775762363458788091, 8.902186558754098285799163241864

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