Properties

Label 2-2151-1.1-c1-0-58
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  − 2.61·2-s + 4.84·4-s − 2.80·5-s + 3.73·7-s − 7.43·8-s + 7.34·10-s + 6.02·11-s − 0.651·13-s − 9.77·14-s + 9.75·16-s − 1.63·17-s − 6.09·19-s − 13.6·20-s − 15.7·22-s − 1.60·23-s + 2.89·25-s + 1.70·26-s + 18.0·28-s − 2.35·29-s − 6.90·31-s − 10.6·32-s + 4.27·34-s − 10.4·35-s − 10.0·37-s + 15.9·38-s + 20.8·40-s − 8.60·41-s + ⋯
L(s)  = 1  − 1.84·2-s + 2.42·4-s − 1.25·5-s + 1.41·7-s − 2.62·8-s + 2.32·10-s + 1.81·11-s − 0.180·13-s − 2.61·14-s + 2.43·16-s − 0.396·17-s − 1.39·19-s − 3.04·20-s − 3.35·22-s − 0.334·23-s + 0.578·25-s + 0.334·26-s + 3.41·28-s − 0.437·29-s − 1.24·31-s − 1.88·32-s + 0.733·34-s − 1.77·35-s − 1.65·37-s + 2.58·38-s + 3.30·40-s − 1.34·41-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 \( 1 + 2.61T + 2T^{2} \)
5 \( 1 + 2.80T + 5T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 - 6.02T + 11T^{2} \)
13 \( 1 + 0.651T + 13T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + 6.90T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 - 8.78T + 43T^{2} \)
47 \( 1 - 6.67T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 - 9.10T + 59T^{2} \)
61 \( 1 + 9.50T + 61T^{2} \)
67 \( 1 + 8.35T + 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 - 8.06T + 73T^{2} \)
79 \( 1 + 4.58T + 79T^{2} \)
83 \( 1 + 9.85T + 83T^{2} \)
89 \( 1 + 18.5T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−8.775925320205634983649485399073, −8.155731663107805779474212352728, −7.26379146285664834941988486092, −6.97723842870121447754751847752, −5.84295121117155168218904955520, −4.40535277079098197790411756359, −3.72136066662112471559870746206, −2.10914586333539963609848940878, −1.37446595942765550490360209042, 0, 1.37446595942765550490360209042, 2.10914586333539963609848940878, 3.72136066662112471559870746206, 4.40535277079098197790411756359, 5.84295121117155168218904955520, 6.97723842870121447754751847752, 7.26379146285664834941988486092, 8.155731663107805779474212352728, 8.775925320205634983649485399073

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