Properties

Label 2-5-1.1-c5-0-0
Degree $2$
Conductor $5$
Sign $1$
Analytic cond. $0.801919$
Root an. cond. $0.895499$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s − 28·4-s + 25·5-s − 8·6-s + 192·7-s − 120·8-s − 227·9-s + 50·10-s − 148·11-s + 112·12-s + 286·13-s + 384·14-s − 100·15-s + 656·16-s − 1.67e3·17-s − 454·18-s + 1.06e3·19-s − 700·20-s − 768·21-s − 296·22-s + 2.97e3·23-s + 480·24-s + 625·25-s + 572·26-s + 1.88e3·27-s − 5.37e3·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.256·3-s − 7/8·4-s + 0.447·5-s − 0.0907·6-s + 1.48·7-s − 0.662·8-s − 0.934·9-s + 0.158·10-s − 0.368·11-s + 0.224·12-s + 0.469·13-s + 0.523·14-s − 0.114·15-s + 0.640·16-s − 1.40·17-s − 0.330·18-s + 0.673·19-s − 0.391·20-s − 0.380·21-s − 0.130·22-s + 1.17·23-s + 0.170·24-s + 1/5·25-s + 0.165·26-s + 0.496·27-s − 1.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $1$
Analytic conductor: \(0.801919\)
Root analytic conductor: \(0.895499\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9710655448\)
\(L(\frac12)\) \(\approx\) \(0.9710655448\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p^{2} T \)
good2 \( 1 - p T + p^{5} T^{2} \)
3 \( 1 + 4 T + p^{5} T^{2} \)
7 \( 1 - 192 T + p^{5} T^{2} \)
11 \( 1 + 148 T + p^{5} T^{2} \)
13 \( 1 - 22 p T + p^{5} T^{2} \)
17 \( 1 + 1678 T + p^{5} T^{2} \)
19 \( 1 - 1060 T + p^{5} T^{2} \)
23 \( 1 - 2976 T + p^{5} T^{2} \)
29 \( 1 + 3410 T + p^{5} T^{2} \)
31 \( 1 + 2448 T + p^{5} T^{2} \)
37 \( 1 - 182 T + p^{5} T^{2} \)
41 \( 1 + 9398 T + p^{5} T^{2} \)
43 \( 1 + 1244 T + p^{5} T^{2} \)
47 \( 1 + 12088 T + p^{5} T^{2} \)
53 \( 1 - 23846 T + p^{5} T^{2} \)
59 \( 1 + 20020 T + p^{5} T^{2} \)
61 \( 1 - 32302 T + p^{5} T^{2} \)
67 \( 1 - 60972 T + p^{5} T^{2} \)
71 \( 1 + 32648 T + p^{5} T^{2} \)
73 \( 1 + 38774 T + p^{5} T^{2} \)
79 \( 1 + 33360 T + p^{5} T^{2} \)
83 \( 1 - 16716 T + p^{5} T^{2} \)
89 \( 1 - 101370 T + p^{5} T^{2} \)
97 \( 1 + 119038 T + p^{5} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.24609425663826686791273389532, −21.93222733105611867420931259922, −20.58783903836707634657406003956, −18.22860474114662304511675025441, −17.33405168388793405888238876888, −14.76134755315793297199518363219, −13.45166559410003997536070100498, −11.25503050513697713183112168572, −8.690118924564440058768821531438, −5.18869999117144960219199272171, 5.18869999117144960219199272171, 8.690118924564440058768821531438, 11.25503050513697713183112168572, 13.45166559410003997536070100498, 14.76134755315793297199518363219, 17.33405168388793405888238876888, 18.22860474114662304511675025441, 20.58783903836707634657406003956, 21.93222733105611867420931259922, 23.24609425663826686791273389532

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