Properties

Label 2-2151-1.1-c1-0-92
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  + 2.15·2-s + 2.64·4-s − 3.41·5-s + 0.201·7-s + 1.39·8-s − 7.35·10-s + 5.35·11-s − 5.86·13-s + 0.434·14-s − 2.28·16-s − 4.05·17-s − 2.42·19-s − 9.03·20-s + 11.5·22-s + 4.58·23-s + 6.65·25-s − 12.6·26-s + 0.533·28-s − 3.44·29-s − 7.68·31-s − 7.71·32-s − 8.73·34-s − 0.687·35-s − 0.781·37-s − 5.23·38-s − 4.76·40-s − 5.87·41-s + ⋯
L(s)  = 1  + 1.52·2-s + 1.32·4-s − 1.52·5-s + 0.0761·7-s + 0.493·8-s − 2.32·10-s + 1.61·11-s − 1.62·13-s + 0.116·14-s − 0.571·16-s − 0.983·17-s − 0.556·19-s − 2.02·20-s + 2.45·22-s + 0.956·23-s + 1.33·25-s − 2.47·26-s + 0.100·28-s − 0.639·29-s − 1.37·31-s − 1.36·32-s − 1.49·34-s − 0.116·35-s − 0.128·37-s − 0.848·38-s − 0.752·40-s − 0.916·41-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 \( 1 - 2.15T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 - 0.201T + 7T^{2} \)
11 \( 1 - 5.35T + 11T^{2} \)
13 \( 1 + 5.86T + 13T^{2} \)
17 \( 1 + 4.05T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 - 4.58T + 23T^{2} \)
29 \( 1 + 3.44T + 29T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 + 0.781T + 37T^{2} \)
41 \( 1 + 5.87T + 41T^{2} \)
43 \( 1 + 8.05T + 43T^{2} \)
47 \( 1 - 3.12T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 9.93T + 59T^{2} \)
61 \( 1 - 5.67T + 61T^{2} \)
67 \( 1 + 6.84T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.765163320801799943532608822268, −7.50488767361613057399097048362, −7.01829872355310655522879987935, −6.36443971894514365336470964910, −5.11008524350250157022591583044, −4.55495908899498687617438815417, −3.87347592667321329959606672035, −3.23825607727718463093019462989, −2.00705383684722046822785918950, 0, 2.00705383684722046822785918950, 3.23825607727718463093019462989, 3.87347592667321329959606672035, 4.55495908899498687617438815417, 5.11008524350250157022591583044, 6.36443971894514365336470964910, 7.01829872355310655522879987935, 7.50488767361613057399097048362, 8.765163320801799943532608822268

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