Properties

Label 4-95e4-1.1-c1e2-0-4
Degree $4$
Conductor $81450625$
Sign $1$
Analytic cond. $5193.36$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s − 2·4-s + 3·6-s − 6·7-s − 3·8-s + 2·9-s − 11-s − 6·12-s − 2·13-s − 6·14-s + 16-s − 6·17-s + 2·18-s − 18·21-s − 22-s − 13·23-s − 9·24-s − 2·26-s − 6·27-s + 12·28-s + 5·29-s + 11·31-s + 2·32-s − 3·33-s − 6·34-s − 4·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 4-s + 1.22·6-s − 2.26·7-s − 1.06·8-s + 2/3·9-s − 0.301·11-s − 1.73·12-s − 0.554·13-s − 1.60·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 3.92·21-s − 0.213·22-s − 2.71·23-s − 1.83·24-s − 0.392·26-s − 1.15·27-s + 2.26·28-s + 0.928·29-s + 1.97·31-s + 0.353·32-s − 0.522·33-s − 1.02·34-s − 2/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81450625\)    =    \(5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(5193.36\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81450625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.417769559\)
\(L(\frac12)\) \(\approx\) \(2.417769559\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_4$ \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 273 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
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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

−7.85798097625417361571397062123, −7.75752718063535588979639526869, −7.36280122613455449797895672376, −6.72708149568712833149822993123, −6.51044743068110102925065968786, −6.01098241864298337147488447676, −5.91928260601735850619997419039, −5.80303546096471051127776450172, −4.77679953874450840308947982058, −4.62016787427499746847381353621, −4.35448518424409690491723181240, −4.06161454083094428445882318802, −3.48524516652856102137724108632, −3.31261115520828932323504285019, −2.95184515086964323631421919604, −2.59139433048905446056033402104, −2.25829524104795198524923758294, −1.93878368770214206404028467889, −0.56791213225024985668349777347, −0.47203532683674249706226830995, 0.47203532683674249706226830995, 0.56791213225024985668349777347, 1.93878368770214206404028467889, 2.25829524104795198524923758294, 2.59139433048905446056033402104, 2.95184515086964323631421919604, 3.31261115520828932323504285019, 3.48524516652856102137724108632, 4.06161454083094428445882318802, 4.35448518424409690491723181240, 4.62016787427499746847381353621, 4.77679953874450840308947982058, 5.80303546096471051127776450172, 5.91928260601735850619997419039, 6.01098241864298337147488447676, 6.51044743068110102925065968786, 6.72708149568712833149822993123, 7.36280122613455449797895672376, 7.75752718063535588979639526869, 7.85798097625417361571397062123

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