Properties

Label 4-1950e2-1.1-c1e2-0-49
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 6·7-s + 4·8-s − 9-s + 4·13-s + 12·14-s + 5·16-s − 2·18-s + 8·26-s + 18·28-s + 10·29-s + 6·32-s − 3·36-s − 4·37-s + 6·47-s + 13·49-s + 12·52-s + 24·56-s + 20·58-s − 26·61-s − 6·63-s + 7·64-s + 26·67-s − 4·72-s − 12·73-s − 8·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 2.26·7-s + 1.41·8-s − 1/3·9-s + 1.10·13-s + 3.20·14-s + 5/4·16-s − 0.471·18-s + 1.56·26-s + 3.40·28-s + 1.85·29-s + 1.06·32-s − 1/2·36-s − 0.657·37-s + 0.875·47-s + 13/7·49-s + 1.66·52-s + 3.20·56-s + 2.62·58-s − 3.32·61-s − 0.755·63-s + 7/8·64-s + 3.17·67-s − 0.471·72-s − 1.40·73-s − 0.929·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.26715829\)
\(L(\frac12)\) \(\approx\) \(10.26715829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.174083056813021041848900294074, −8.900708294718218593908172421737, −8.362753038381910379653896399934, −8.221386122725656731195566504438, −7.84962674250234991788385617270, −7.33189067866850681642682591467, −7.05557250673826687196833987821, −6.41317365972876743535271740190, −5.95174744379117662786101475525, −5.91875119374335620317214570393, −5.05038657882760264012725781375, −4.98319389822163979942693905356, −4.53750356151393784038598312343, −4.27924155719094403777257246368, −3.44662618665729491112720039250, −3.35178702918451943111584424923, −2.49153215459591138965778677445, −2.09336847200769477098002337447, −1.45491289552526550486262302402, −1.02595622375540356416283476511, 1.02595622375540356416283476511, 1.45491289552526550486262302402, 2.09336847200769477098002337447, 2.49153215459591138965778677445, 3.35178702918451943111584424923, 3.44662618665729491112720039250, 4.27924155719094403777257246368, 4.53750356151393784038598312343, 4.98319389822163979942693905356, 5.05038657882760264012725781375, 5.91875119374335620317214570393, 5.95174744379117662786101475525, 6.41317365972876743535271740190, 7.05557250673826687196833987821, 7.33189067866850681642682591467, 7.84962674250234991788385617270, 8.221386122725656731195566504438, 8.362753038381910379653896399934, 8.900708294718218593908172421737, 9.174083056813021041848900294074

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