Properties

Label 2-58835-1.1-c1-0-2
Degree $2$
Conductor $58835$
Sign $-1$
Analytic cond. $469.799$
Root an. cond. $21.6748$
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  − 3-s − 2·4-s − 5-s − 7-s − 2·9-s + 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s − 3·17-s − 2·19-s + 2·20-s + 21-s − 6·23-s + 25-s + 5·27-s + 2·28-s − 3·29-s − 4·31-s − 3·33-s + 35-s + 4·36-s + 2·37-s + 5·39-s − 10·43-s − 6·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s + 0.800·39-s − 1.52·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58835\)    =    \(5 \cdot 7 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(469.799\)
Root analytic conductor: \(21.6748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58835} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58835,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
41 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.61478340392634, −14.25230170280462, −13.48455125280384, −13.07714754018952, −12.39348175761571, −12.20611487024252, −11.50177033161236, −11.23463737110482, −10.41887726582497, −9.935393917216482, −9.410274877837081, −9.044822416844772, −8.310965598426057, −8.002434224761507, −7.255192509437189, −6.549156683494704, −6.220290782772069, −5.477106695543866, −4.851488399365523, −4.571689625293214, −3.707354031760159, −3.405672214752647, −2.408496029809493, −1.682361435923316, −0.4955358127107946, 0, 0.4955358127107946, 1.682361435923316, 2.408496029809493, 3.405672214752647, 3.707354031760159, 4.571689625293214, 4.851488399365523, 5.477106695543866, 6.220290782772069, 6.549156683494704, 7.255192509437189, 8.002434224761507, 8.310965598426057, 9.044822416844772, 9.410274877837081, 9.935393917216482, 10.41887726582497, 11.23463737110482, 11.50177033161236, 12.20611487024252, 12.39348175761571, 13.07714754018952, 13.48455125280384, 14.25230170280462, 14.61478340392634

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