Properties

Label 2-2151-1.1-c1-0-47
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.65·2-s + 5.03·4-s − 3.62·5-s − 2.98·7-s + 8.05·8-s − 9.60·10-s + 1.77·11-s + 5.96·13-s − 7.91·14-s + 11.2·16-s + 5.47·17-s + 5.48·19-s − 18.2·20-s + 4.72·22-s − 2.11·23-s + 8.11·25-s + 15.8·26-s − 15.0·28-s − 2.88·29-s + 2.48·31-s + 13.8·32-s + 14.5·34-s + 10.8·35-s + 5.90·37-s + 14.5·38-s − 29.1·40-s + 2.24·41-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.51·4-s − 1.61·5-s − 1.12·7-s + 2.84·8-s − 3.03·10-s + 0.536·11-s + 1.65·13-s − 2.11·14-s + 2.82·16-s + 1.32·17-s + 1.25·19-s − 4.07·20-s + 1.00·22-s − 0.441·23-s + 1.62·25-s + 3.10·26-s − 2.83·28-s − 0.535·29-s + 0.445·31-s + 2.44·32-s + 2.48·34-s + 1.82·35-s + 0.970·37-s + 2.36·38-s − 4.61·40-s + 0.349·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.637061732\)
\(L(\frac12)\) \(\approx\) \(4.637061732\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - 2.65T + 2T^{2} \)
5 \( 1 + 3.62T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 - 1.77T + 11T^{2} \)
13 \( 1 - 5.96T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 + 2.11T + 23T^{2} \)
29 \( 1 + 2.88T + 29T^{2} \)
31 \( 1 - 2.48T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 4.83T + 59T^{2} \)
61 \( 1 - 0.954T + 61T^{2} \)
67 \( 1 + 4.22T + 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + 4.78T + 73T^{2} \)
79 \( 1 - 2.29T + 79T^{2} \)
83 \( 1 - 6.12T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 4.74T + 97T^{2} \)
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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

−9.013379712369381545310888876694, −7.74090915769233169104812180691, −7.50388159499122449336835681224, −6.28770825030048881813724462263, −6.05959889937136392211757443179, −4.88447829638734184787990485490, −3.94129943463316871707341075116, −3.49556610480522950858407703241, −3.03645639769226996550125614583, −1.16156701404506241434678792508, 1.16156701404506241434678792508, 3.03645639769226996550125614583, 3.49556610480522950858407703241, 3.94129943463316871707341075116, 4.88447829638734184787990485490, 6.05959889937136392211757443179, 6.28770825030048881813724462263, 7.50388159499122449336835681224, 7.74090915769233169104812180691, 9.013379712369381545310888876694

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