Properties

Label 2-2151-1.1-c1-0-86
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  − 0.854·2-s − 1.26·4-s + 2.82·5-s + 3.33·7-s + 2.79·8-s − 2.41·10-s + 0.210·11-s − 4.81·13-s − 2.84·14-s + 0.150·16-s − 4.49·17-s − 4.63·19-s − 3.58·20-s − 0.179·22-s − 7.53·23-s + 2.95·25-s + 4.11·26-s − 4.22·28-s − 2.29·29-s − 7.14·31-s − 5.71·32-s + 3.83·34-s + 9.39·35-s − 1.94·37-s + 3.96·38-s + 7.88·40-s − 8.56·41-s + ⋯
L(s)  = 1  − 0.604·2-s − 0.634·4-s + 1.26·5-s + 1.25·7-s + 0.987·8-s − 0.762·10-s + 0.0634·11-s − 1.33·13-s − 0.760·14-s + 0.0376·16-s − 1.08·17-s − 1.06·19-s − 0.800·20-s − 0.0383·22-s − 1.57·23-s + 0.590·25-s + 0.807·26-s − 0.799·28-s − 0.425·29-s − 1.28·31-s − 1.01·32-s + 0.658·34-s + 1.58·35-s − 0.320·37-s + 0.642·38-s + 1.24·40-s − 1.33·41-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 \( 1 + 0.854T + 2T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 3.33T + 7T^{2} \)
11 \( 1 - 0.210T + 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + 4.63T + 19T^{2} \)
23 \( 1 + 7.53T + 23T^{2} \)
29 \( 1 + 2.29T + 29T^{2} \)
31 \( 1 + 7.14T + 31T^{2} \)
37 \( 1 + 1.94T + 37T^{2} \)
41 \( 1 + 8.56T + 41T^{2} \)
43 \( 1 + 1.06T + 43T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 + 4.11T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 8.53T + 61T^{2} \)
67 \( 1 - 16.0T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 2.49T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 4.26T + 89T^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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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

−8.664212883178060916561153919961, −8.172911836129150831156603771631, −7.30125079740309602653101806087, −6.36911817869845580652154337875, −5.30416378973764711747671953197, −4.84375272352294101252926796877, −3.94419495073796829462172498568, −2.03909662811114627247668496976, −1.88378557177365249338484410473, 0, 1.88378557177365249338484410473, 2.03909662811114627247668496976, 3.94419495073796829462172498568, 4.84375272352294101252926796877, 5.30416378973764711747671953197, 6.36911817869845580652154337875, 7.30125079740309602653101806087, 8.172911836129150831156603771631, 8.664212883178060916561153919961

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