Properties

Label 2-4368-1.1-c1-0-31
Degree $2$
Conductor $4368$
Sign $1$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 13-s + 2·15-s + 2·17-s − 4·19-s − 21-s + 4·23-s − 25-s + 27-s + 10·29-s − 8·31-s + 4·33-s − 2·35-s + 2·37-s − 39-s + 10·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.080426503\)
\(L(\frac12)\) \(\approx\) \(3.080426503\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.591909054463627013254143334549, −7.60737624627539231014495398201, −6.83423409318481332335160340857, −6.27123411237532440751009381950, −5.52313078018151925153558509277, −4.51035263133546415298573729927, −3.76910138393662928748214031195, −2.82842048602009961166929070702, −2.02297835901821089070845609831, −1.01235796598496708925518877535, 1.01235796598496708925518877535, 2.02297835901821089070845609831, 2.82842048602009961166929070702, 3.76910138393662928748214031195, 4.51035263133546415298573729927, 5.52313078018151925153558509277, 6.27123411237532440751009381950, 6.83423409318481332335160340857, 7.60737624627539231014495398201, 8.591909054463627013254143334549

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