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 − 16·19-s + 18·29-s + 2·31-s − 22·41-s − 4·44-s − 49-s + 14·59-s − 10·61-s − 64-s + 8·71-s + 16·76-s + 12·79-s − 20·89-s − 8·109-s − 18·116-s − 10·121-s − 2·124-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 − 3.67·19-s + 3.34·29-s + 0.359·31-s − 3.43·41-s − 0.603·44-s − 1/7·49-s + 1.82·59-s − 1.28·61-s − 1/8·64-s + 0.949·71-s + 1.83·76-s + 1.35·79-s − 2.11·89-s − 0.766·109-s − 1.67·116-s − 0.909·121-s − 0.179·124-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.158524722$
$L(\frac12)$  $\approx$  $1.158524722$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
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

−8.853275170807142537891153385401, −8.359063473281505591935829106303, −8.340033520648519684455229869048, −8.101684733450376567932985661289, −7.25166980335507169927531130144, −6.73277316906065924708901819067, −6.61436164253623202977620184795, −6.32957002532125918216990431556, −6.14202258076991458656565764398, −5.19463959774859308182043047669, −5.04970164333507718777609582642, −4.47297915487857364706196279247, −4.32464660042979915556858373212, −3.76511834172431116782694093027, −3.51648091045218432085558216112, −2.62588015808960914988669655533, −2.46935528906122070996103266726, −1.67568919804336700782523855238, −1.26326178822355029715871554354, −0.34920238595241510993526294456, 0.34920238595241510993526294456, 1.26326178822355029715871554354, 1.67568919804336700782523855238, 2.46935528906122070996103266726, 2.62588015808960914988669655533, 3.51648091045218432085558216112, 3.76511834172431116782694093027, 4.32464660042979915556858373212, 4.47297915487857364706196279247, 5.04970164333507718777609582642, 5.19463959774859308182043047669, 6.14202258076991458656565764398, 6.32957002532125918216990431556, 6.61436164253623202977620184795, 6.73277316906065924708901819067, 7.25166980335507169927531130144, 8.101684733450376567932985661289, 8.340033520648519684455229869048, 8.359063473281505591935829106303, 8.853275170807142537891153385401

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