Properties

Label 2-58-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s − 2·9-s + 10-s − 3·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s + 8·17-s − 2·18-s + 20-s + 2·21-s − 3·22-s + 4·23-s − 24-s − 4·25-s − 26-s + 5·27-s − 2·28-s − 29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s + 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.962·27-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: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.033207509\)
\(L(\frac12)\) \(\approx\) \(1.033207509\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.09322040009074974335092601729, −13.99719858770620348924737003173, −12.92256133309152763022923630908, −12.00978753036483029980569259298, −10.74800585012115946663581328742, −9.633495492381883994739439375568, −7.71436185563727090632996745083, −6.11658999858863662469236119804, −5.25136790623537531107690201816, −3.07071881135695746551942332895, 3.07071881135695746551942332895, 5.25136790623537531107690201816, 6.11658999858863662469236119804, 7.71436185563727090632996745083, 9.633495492381883994739439375568, 10.74800585012115946663581328742, 12.00978753036483029980569259298, 12.92256133309152763022923630908, 13.99719858770620348924737003173, 15.09322040009074974335092601729

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