Properties

Label 4-26e2-1.1-c1e2-0-3
Degree $4$
Conductor $676$
Sign $1$
Analytic cond. $0.0431023$
Root an. cond. $0.455643$
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  − 2·2-s + 2·3-s + 3·4-s − 6·5-s − 4·6-s − 2·7-s − 4·8-s − 3·9-s + 12·10-s + 12·11-s + 6·12-s + 2·13-s + 4·14-s − 12·15-s + 5·16-s − 6·17-s + 6·18-s + 4·19-s − 18·20-s − 4·21-s − 24·22-s − 8·24-s + 17·25-s − 4·26-s − 14·27-s − 6·28-s + 12·29-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s − 9-s + 3.79·10-s + 3.61·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s − 3.09·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 0.917·19-s − 4.02·20-s − 0.872·21-s − 5.11·22-s − 1.63·24-s + 17/5·25-s − 0.784·26-s − 2.69·27-s − 1.13·28-s + 2.22·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0431023\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2658192833\)
\(L(\frac12)\) \(\approx\) \(0.2658192833\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−19.6933997517, −19.6933997517, −19.1444693222, −19.1444693222, −17.5187720040, −17.5187720040, −16.3289342707, −16.3289342707, −15.2621223361, −15.2621223361, −14.1081230548, −14.1081230548, −12.1081086828, −12.1081086828, −11.2148699926, −11.2148699926, −9.25208611998, −9.25208611998, −8.28912505160, −8.28912505160, −6.76423132328, −6.76423132328, −3.64028761626, −3.64028761626, 3.64028761626, 3.64028761626, 6.76423132328, 6.76423132328, 8.28912505160, 8.28912505160, 9.25208611998, 9.25208611998, 11.2148699926, 11.2148699926, 12.1081086828, 12.1081086828, 14.1081230548, 14.1081230548, 15.2621223361, 15.2621223361, 16.3289342707, 16.3289342707, 17.5187720040, 17.5187720040, 19.1444693222, 19.1444693222, 19.6933997517, 19.6933997517

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