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 + 10·11-s + 16-s + 6·19-s − 12·29-s − 8·31-s − 22·41-s − 10·44-s − 49-s + 8·59-s − 4·61-s − 64-s + 20·71-s − 6·76-s + 4·79-s − 22·89-s + 36·109-s + 12·116-s + 53·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s + 3.01·11-s + 1/4·16-s + 1.37·19-s − 2.22·29-s − 1.43·31-s − 3.43·41-s − 1.50·44-s − 1/7·49-s + 1.04·59-s − 0.512·61-s − 1/8·64-s + 2.37·71-s − 0.688·76-s + 0.450·79-s − 2.33·89-s + 3.44·109-s + 1.11·116-s + 4.81·121-s + 0.718·124-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.456992033$
$L(\frac12)$  $\approx$  $2.456992033$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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

−8.940999227549445112879363638634, −8.647550035908286158871184300160, −8.193752853026428756146186289973, −7.76603679378902413954779017383, −7.18559469412112112694076262632, −7.04153297606879144723029512768, −6.67356107790910299249991892052, −6.35628908001268040720223029535, −5.63126177784664005361074925686, −5.56940118472467535869881663402, −5.08876612674051165008006426965, −4.57512033006114539987923451965, −3.97859582970042917261455859819, −3.82591323704486498966740123322, −3.29831504988979286938665182240, −3.27044647177164368629895254730, −1.89161967619458329935438229526, −1.85702953716993131741810715062, −1.26151310040410465549325609124, −0.52581178960635267967458547310, 0.52581178960635267967458547310, 1.26151310040410465549325609124, 1.85702953716993131741810715062, 1.89161967619458329935438229526, 3.27044647177164368629895254730, 3.29831504988979286938665182240, 3.82591323704486498966740123322, 3.97859582970042917261455859819, 4.57512033006114539987923451965, 5.08876612674051165008006426965, 5.56940118472467535869881663402, 5.63126177784664005361074925686, 6.35628908001268040720223029535, 6.67356107790910299249991892052, 7.04153297606879144723029512768, 7.18559469412112112694076262632, 7.76603679378902413954779017383, 8.193752853026428756146186289973, 8.647550035908286158871184300160, 8.940999227549445112879363638634

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