Properties

Label 2-58-1.1-c1-0-1
Degree $2$
Conductor $58$
Sign $-1$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
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 − 3·3-s + 4-s − 3·5-s + 3·6-s − 2·7-s − 8-s + 6·9-s + 3·10-s − 11-s − 3·12-s + 3·13-s + 2·14-s + 9·15-s + 16-s − 4·17-s − 6·18-s − 8·19-s − 3·20-s + 6·21-s + 22-s + 3·24-s + 4·25-s − 3·26-s − 9·27-s − 2·28-s − 29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 0.301·11-s − 0.866·12-s + 0.832·13-s + 0.534·14-s + 2.32·15-s + 1/4·16-s − 0.970·17-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 1.30·21-s + 0.213·22-s + 0.612·24-s + 4/5·25-s − 0.588·26-s − 1.73·27-s − 0.377·28-s − 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $-1$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
29 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.47776366797874230167954280335, −12.96417955859963649393323437145, −12.07048630043891833907177728325, −11.08257082663040259358467539040, −10.46054334513251190747035603866, −8.643506374817898327978246248838, −7.09229495493052892657316288498, −6.11755064298029935226028768008, −4.22146327216160305586457647308, 0, 4.22146327216160305586457647308, 6.11755064298029935226028768008, 7.09229495493052892657316288498, 8.643506374817898327978246248838, 10.46054334513251190747035603866, 11.08257082663040259358467539040, 12.07048630043891833907177728325, 12.96417955859963649393323437145, 15.47776366797874230167954280335

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