Properties

Label 2-2151-1.1-c1-0-81
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.74·2-s + 5.54·4-s + 2.02·5-s − 1.84·7-s + 9.74·8-s + 5.56·10-s − 0.918·11-s + 4.45·13-s − 5.06·14-s + 15.6·16-s + 1.57·17-s − 5.99·19-s + 11.2·20-s − 2.52·22-s + 2.16·23-s − 0.903·25-s + 12.2·26-s − 10.2·28-s − 4.69·29-s − 4.00·31-s + 23.5·32-s + 4.33·34-s − 3.73·35-s − 3.02·37-s − 16.4·38-s + 19.7·40-s − 9.24·41-s + ⋯
L(s)  = 1  + 1.94·2-s + 2.77·4-s + 0.905·5-s − 0.696·7-s + 3.44·8-s + 1.75·10-s − 0.276·11-s + 1.23·13-s − 1.35·14-s + 3.92·16-s + 0.382·17-s − 1.37·19-s + 2.51·20-s − 0.538·22-s + 0.452·23-s − 0.180·25-s + 2.39·26-s − 1.93·28-s − 0.872·29-s − 0.718·31-s + 4.17·32-s + 0.742·34-s − 0.630·35-s − 0.497·37-s − 2.67·38-s + 3.11·40-s − 1.44·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.027325453\)
\(L(\frac12)\) \(\approx\) \(7.027325453\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 2T^{2} \)
5 \( 1 - 2.02T + 5T^{2} \)
7 \( 1 + 1.84T + 7T^{2} \)
11 \( 1 + 0.918T + 11T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
17 \( 1 - 1.57T + 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 + 4.00T + 31T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 + 9.24T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 - 6.23T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 1.98T + 61T^{2} \)
67 \( 1 + 3.42T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 4.04T + 73T^{2} \)
79 \( 1 - 1.44T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−9.118025057722467395158089875577, −8.101004373169434491715494287240, −6.93029401736591212995023870169, −6.51079172242930976447360898331, −5.67724955741845063875001293975, −5.31981197866294783484317708105, −4.04832505502097455606565070003, −3.52327690296556952902121393405, −2.48327703585472300189257214203, −1.64597254419736338759588049720, 1.64597254419736338759588049720, 2.48327703585472300189257214203, 3.52327690296556952902121393405, 4.04832505502097455606565070003, 5.31981197866294783484317708105, 5.67724955741845063875001293975, 6.51079172242930976447360898331, 6.93029401736591212995023870169, 8.101004373169434491715494287240, 9.118025057722467395158089875577

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