Properties

Label 2-2151-1.1-c3-0-283
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $126.913$
Root an. cond. $11.2655$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.15·2-s + 18.5·4-s − 1.27·5-s − 23.0·7-s + 54.4·8-s − 6.55·10-s + 4.21·11-s − 16.3·13-s − 118.·14-s + 132.·16-s + 98.0·17-s − 60.2·19-s − 23.6·20-s + 21.7·22-s − 168.·23-s − 123.·25-s − 84.4·26-s − 428.·28-s − 72.3·29-s − 26.8·31-s + 245.·32-s + 505.·34-s + 29.3·35-s − 86.2·37-s − 310.·38-s − 69.2·40-s + 137.·41-s + ⋯
L(s)  = 1  + 1.82·2-s + 2.32·4-s − 0.113·5-s − 1.24·7-s + 2.40·8-s − 0.207·10-s + 0.115·11-s − 0.349·13-s − 2.27·14-s + 2.06·16-s + 1.39·17-s − 0.727·19-s − 0.263·20-s + 0.210·22-s − 1.53·23-s − 0.987·25-s − 0.636·26-s − 2.89·28-s − 0.463·29-s − 0.155·31-s + 1.35·32-s + 2.54·34-s + 0.141·35-s − 0.383·37-s − 1.32·38-s − 0.273·40-s + 0.523·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(126.913\)
Root analytic conductor: \(11.2655\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 239T \)
good2 \( 1 - 5.15T + 8T^{2} \)
5 \( 1 + 1.27T + 125T^{2} \)
7 \( 1 + 23.0T + 343T^{2} \)
11 \( 1 - 4.21T + 1.33e3T^{2} \)
13 \( 1 + 16.3T + 2.19e3T^{2} \)
17 \( 1 - 98.0T + 4.91e3T^{2} \)
19 \( 1 + 60.2T + 6.85e3T^{2} \)
23 \( 1 + 168.T + 1.21e4T^{2} \)
29 \( 1 + 72.3T + 2.43e4T^{2} \)
31 \( 1 + 26.8T + 2.97e4T^{2} \)
37 \( 1 + 86.2T + 5.06e4T^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + 406.T + 7.95e4T^{2} \)
47 \( 1 - 382.T + 1.03e5T^{2} \)
53 \( 1 - 173.T + 1.48e5T^{2} \)
59 \( 1 + 425.T + 2.05e5T^{2} \)
61 \( 1 + 508.T + 2.26e5T^{2} \)
67 \( 1 + 650.T + 3.00e5T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 + 291.T + 3.89e5T^{2} \)
79 \( 1 - 948.T + 4.93e5T^{2} \)
83 \( 1 - 585.T + 5.71e5T^{2} \)
89 \( 1 - 587.T + 7.04e5T^{2} \)
97 \( 1 + 479.T + 9.12e5T^{2} \)
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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

−7.938280040407251678876543323118, −7.31001255608151512635566996752, −6.32624007893006791524079235437, −5.97848175353302801782346190939, −5.15113382238317044338766633387, −4.06459724406256727867027800233, −3.59734723508553439892904594190, −2.75719076586169883158941080293, −1.76763408802928591520211316820, 0, 1.76763408802928591520211316820, 2.75719076586169883158941080293, 3.59734723508553439892904594190, 4.06459724406256727867027800233, 5.15113382238317044338766633387, 5.97848175353302801782346190939, 6.32624007893006791524079235437, 7.31001255608151512635566996752, 7.938280040407251678876543323118

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