Properties

Label 4-5171-1.1-c1e2-0-0
Degree $4$
Conductor $5171$
Sign $-1$
Analytic cond. $0.329707$
Root an. cond. $0.757760$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3·5-s + 6-s − 3·7-s − 3·8-s + 3·9-s + 3·10-s − 8·11-s − 12-s + 4·13-s + 3·14-s + 3·15-s + 16-s − 2·17-s − 3·18-s + 2·19-s − 3·20-s + 3·21-s + 8·22-s + 23-s + 3·24-s + 25-s − 4·26-s − 8·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s + 9-s + 0.948·10-s − 2.41·11-s − 0.288·12-s + 1.10·13-s + 0.801·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.458·19-s − 0.670·20-s + 0.654·21-s + 1.70·22-s + 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.784·26-s − 1.53·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5171 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5171 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5171\)
Sign: $-1$
Analytic conductor: \(0.329707\)
Root analytic conductor: \(0.757760\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5171,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad5171$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 72 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T - 18 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 86 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 5 T - 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 11 T + 76 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−17.8800484528, −17.3051648183, −16.4624280124, −15.9888141621, −15.6972004576, −15.5397275659, −15.2917950317, −13.9584397857, −13.3770057643, −12.8293752742, −12.4850102158, −11.8879015351, −11.1199101188, −10.8476495142, −10.2953456302, −9.51690465138, −9.07402619722, −7.99658124104, −7.86011292265, −7.06926639476, −6.33852104790, −5.68760035268, −4.65694256324, −3.63162630126, −2.75959377161, 0, 2.75959377161, 3.63162630126, 4.65694256324, 5.68760035268, 6.33852104790, 7.06926639476, 7.86011292265, 7.99658124104, 9.07402619722, 9.51690465138, 10.2953456302, 10.8476495142, 11.1199101188, 11.8879015351, 12.4850102158, 12.8293752742, 13.3770057643, 13.9584397857, 15.2917950317, 15.5397275659, 15.6972004576, 15.9888141621, 16.4624280124, 17.3051648183, 17.8800484528

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