Properties

Label 2-2151-1.1-c1-0-88
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.38·2-s − 0.0912·4-s + 1.35·5-s − 1.50·7-s − 2.88·8-s + 1.86·10-s + 5.20·11-s − 6.03·13-s − 2.07·14-s − 3.80·16-s − 2.23·17-s − 5.35·19-s − 0.123·20-s + 7.19·22-s − 2.95·23-s − 3.17·25-s − 8.34·26-s + 0.137·28-s + 0.529·29-s − 3.72·31-s + 0.515·32-s − 3.08·34-s − 2.03·35-s + 5.79·37-s − 7.40·38-s − 3.90·40-s + 4.00·41-s + ⋯
L(s)  = 1  + 0.976·2-s − 0.0456·4-s + 0.604·5-s − 0.568·7-s − 1.02·8-s + 0.590·10-s + 1.57·11-s − 1.67·13-s − 0.555·14-s − 0.952·16-s − 0.542·17-s − 1.22·19-s − 0.0275·20-s + 1.53·22-s − 0.616·23-s − 0.634·25-s − 1.63·26-s + 0.0259·28-s + 0.0982·29-s − 0.668·31-s + 0.0911·32-s − 0.529·34-s − 0.343·35-s + 0.952·37-s − 1.20·38-s − 0.617·40-s + 0.624·41-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: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - 1.38T + 2T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
7 \( 1 + 1.50T + 7T^{2} \)
11 \( 1 - 5.20T + 11T^{2} \)
13 \( 1 + 6.03T + 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 + 5.35T + 19T^{2} \)
23 \( 1 + 2.95T + 23T^{2} \)
29 \( 1 - 0.529T + 29T^{2} \)
31 \( 1 + 3.72T + 31T^{2} \)
37 \( 1 - 5.79T + 37T^{2} \)
41 \( 1 - 4.00T + 41T^{2} \)
43 \( 1 - 4.50T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 8.22T + 53T^{2} \)
59 \( 1 + 5.35T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 + 9.35T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 5.72T + 79T^{2} \)
83 \( 1 - 4.70T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.980268442854726701948695581515, −7.84433607754896367213430927799, −6.67202764679825872444788152746, −6.32543110919450094616692261084, −5.48880752230952095578416701523, −4.46324747300008663848214555266, −4.02115766674652948604696470076, −2.86706043048514136773633372796, −1.93739781660191230361709662110, 0, 1.93739781660191230361709662110, 2.86706043048514136773633372796, 4.02115766674652948604696470076, 4.46324747300008663848214555266, 5.48880752230952095578416701523, 6.32543110919450094616692261084, 6.67202764679825872444788152746, 7.84433607754896367213430927799, 8.980268442854726701948695581515

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