Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \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  − 9-s + 2·11-s + 8·19-s + 18·29-s − 4·31-s + 16·41-s − 49-s − 8·59-s + 8·61-s + 10·71-s + 30·79-s + 81-s − 16·89-s − 2·99-s + 12·101-s − 2·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 0.603·11-s + 1.83·19-s + 3.34·29-s − 0.718·31-s + 2.49·41-s − 1/7·49-s − 1.04·59-s + 1.02·61-s + 1.18·71-s + 3.37·79-s + 1/9·81-s − 1.69·89-s − 0.201·99-s + 1.19·101-s − 0.191·109-s − 1.72·121-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 + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4410000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2100} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 4410000,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.284824893\)
\(L(\frac12)\)  \(\approx\)  \(3.284824893\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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

−9.220977692434289125813218020240, −9.146465870184735332934552115827, −8.351351856227891143231312056968, −8.188741486731160862230965141119, −7.86820580502641555615456375265, −7.32299928303650050548369993068, −6.90661490351405145622428119943, −6.66280959340880125256554989821, −5.97022906938966719531904234320, −5.96931559488246018620684285268, −5.10506954473409917021379668141, −5.08580509834219888984809235677, −4.35614797214547920585632639742, −4.10716289327802357148768155796, −3.27550462808842902712028244703, −3.16746886683307598405410565243, −2.52870192577483275079273184112, −1.99243152912656953148516874951, −0.926719901077701485248323616985, −0.907060684603556304485651327637, 0.907060684603556304485651327637, 0.926719901077701485248323616985, 1.99243152912656953148516874951, 2.52870192577483275079273184112, 3.16746886683307598405410565243, 3.27550462808842902712028244703, 4.10716289327802357148768155796, 4.35614797214547920585632639742, 5.08580509834219888984809235677, 5.10506954473409917021379668141, 5.96931559488246018620684285268, 5.97022906938966719531904234320, 6.66280959340880125256554989821, 6.90661490351405145622428119943, 7.32299928303650050548369993068, 7.86820580502641555615456375265, 8.188741486731160862230965141119, 8.351351856227891143231312056968, 9.146465870184735332934552115827, 9.220977692434289125813218020240

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