Properties

Label 2-2151-1.1-c1-0-63
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  − 1.66·2-s + 0.787·4-s + 3.19·5-s − 2.48·7-s + 2.02·8-s − 5.33·10-s − 4.44·11-s − 0.0955·13-s + 4.15·14-s − 4.95·16-s + 0.658·17-s + 5.40·19-s + 2.51·20-s + 7.42·22-s + 0.635·23-s + 5.21·25-s + 0.159·26-s − 1.95·28-s − 9.38·29-s + 7.21·31-s + 4.22·32-s − 1.09·34-s − 7.94·35-s − 11.9·37-s − 9.03·38-s + 6.46·40-s − 6.90·41-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.393·4-s + 1.42·5-s − 0.939·7-s + 0.715·8-s − 1.68·10-s − 1.34·11-s − 0.0265·13-s + 1.10·14-s − 1.23·16-s + 0.159·17-s + 1.24·19-s + 0.562·20-s + 1.58·22-s + 0.132·23-s + 1.04·25-s + 0.0312·26-s − 0.369·28-s − 1.74·29-s + 1.29·31-s + 0.746·32-s − 0.188·34-s − 1.34·35-s − 1.96·37-s − 1.46·38-s + 1.02·40-s − 1.07·41-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 \( 1 + 1.66T + 2T^{2} \)
5 \( 1 - 3.19T + 5T^{2} \)
7 \( 1 + 2.48T + 7T^{2} \)
11 \( 1 + 4.44T + 11T^{2} \)
13 \( 1 + 0.0955T + 13T^{2} \)
17 \( 1 - 0.658T + 17T^{2} \)
19 \( 1 - 5.40T + 19T^{2} \)
23 \( 1 - 0.635T + 23T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 - 7.21T + 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 + 6.90T + 41T^{2} \)
43 \( 1 - 7.01T + 43T^{2} \)
47 \( 1 + 7.53T + 47T^{2} \)
53 \( 1 + 0.341T + 53T^{2} \)
59 \( 1 + 1.57T + 59T^{2} \)
61 \( 1 + 1.38T + 61T^{2} \)
67 \( 1 + 0.142T + 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 - 0.402T + 73T^{2} \)
79 \( 1 - 6.98T + 79T^{2} \)
83 \( 1 + 0.0483T + 83T^{2} \)
89 \( 1 + 5.95T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.927870691412319687735559983890, −8.029586570724229026803615897883, −7.28385153200327639755752272446, −6.49826894850728362870871502168, −5.52589298918300865401802394553, −5.00493994430224528986221802919, −3.41415297775985629112846724898, −2.43606746270701686907622283431, −1.45000844540558704212937234626, 0, 1.45000844540558704212937234626, 2.43606746270701686907622283431, 3.41415297775985629112846724898, 5.00493994430224528986221802919, 5.52589298918300865401802394553, 6.49826894850728362870871502168, 7.28385153200327639755752272446, 8.029586570724229026803615897883, 8.927870691412319687735559983890

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