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 − 6·11-s + 16-s + 14·19-s − 12·29-s − 8·31-s + 18·41-s + 6·44-s − 49-s + 24·59-s − 20·61-s − 64-s − 12·71-s − 14·76-s − 28·79-s − 30·89-s − 28·109-s + 12·116-s + 5·121-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 − 1.80·11-s + 1/4·16-s + 3.21·19-s − 2.22·29-s − 1.43·31-s + 2.81·41-s + 0.904·44-s − 1/7·49-s + 3.12·59-s − 2.56·61-s − 1/8·64-s − 1.42·71-s − 1.60·76-s − 3.15·79-s − 3.17·89-s − 2.68·109-s + 1.11·116-s + 5/11·121-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9625451271$
$L(\frac12)$  $\approx$  $0.9625451271$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
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

−9.036394571262452559748210518960, −8.422744676750707553716049824440, −8.001046879665573109320359018896, −7.68023298155020637923512157752, −7.50317155980427810880870642999, −7.07982932957253178256932905795, −6.94419783847639042748313663024, −5.78708378126927330382522670052, −5.67219923136630603838785016438, −5.46866095553103841793803729044, −5.39430893478400508669661365282, −4.44876803856771021257999655318, −4.37657507233285635145635599896, −3.68302071037248763598897225823, −3.28397378274944978768731934921, −2.78582662491447571525563785209, −2.55444404414442935285995910406, −1.65270176310480196297829155039, −1.21157222186398183894077341966, −0.32523492560136225421712528518, 0.32523492560136225421712528518, 1.21157222186398183894077341966, 1.65270176310480196297829155039, 2.55444404414442935285995910406, 2.78582662491447571525563785209, 3.28397378274944978768731934921, 3.68302071037248763598897225823, 4.37657507233285635145635599896, 4.44876803856771021257999655318, 5.39430893478400508669661365282, 5.46866095553103841793803729044, 5.67219923136630603838785016438, 5.78708378126927330382522670052, 6.94419783847639042748313663024, 7.07982932957253178256932905795, 7.50317155980427810880870642999, 7.68023298155020637923512157752, 8.001046879665573109320359018896, 8.422744676750707553716049824440, 9.036394571262452559748210518960

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