Properties

Label 2-1859-1.1-c1-0-113
Degree $2$
Conductor $1859$
Sign $1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
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  + 2.56·2-s + 1.12·3-s + 4.57·4-s + 1.24·5-s + 2.88·6-s + 2.50·7-s + 6.60·8-s − 1.73·9-s + 3.20·10-s + 11-s + 5.15·12-s + 6.42·14-s + 1.40·15-s + 7.78·16-s + 2.78·17-s − 4.43·18-s − 5.79·19-s + 5.70·20-s + 2.82·21-s + 2.56·22-s − 8.60·23-s + 7.44·24-s − 3.44·25-s − 5.33·27-s + 11.4·28-s − 0.584·29-s + 3.60·30-s + ⋯
L(s)  = 1  + 1.81·2-s + 0.650·3-s + 2.28·4-s + 0.558·5-s + 1.17·6-s + 0.946·7-s + 2.33·8-s − 0.576·9-s + 1.01·10-s + 0.301·11-s + 1.48·12-s + 1.71·14-s + 0.363·15-s + 1.94·16-s + 0.676·17-s − 1.04·18-s − 1.32·19-s + 1.27·20-s + 0.615·21-s + 0.546·22-s − 1.79·23-s + 1.51·24-s − 0.688·25-s − 1.02·27-s + 2.16·28-s − 0.108·29-s + 0.658·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(14.8441\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.291850241\)
\(L(\frac12)\) \(\approx\) \(7.291850241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 2.56T + 2T^{2} \)
3 \( 1 - 1.12T + 3T^{2} \)
5 \( 1 - 1.24T + 5T^{2} \)
7 \( 1 - 2.50T + 7T^{2} \)
17 \( 1 - 2.78T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
23 \( 1 + 8.60T + 23T^{2} \)
29 \( 1 + 0.584T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 - 8.21T + 37T^{2} \)
41 \( 1 - 0.167T + 41T^{2} \)
43 \( 1 - 9.05T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 1.39T + 53T^{2} \)
59 \( 1 - 5.61T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 - 7.96T + 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 9.73T + 83T^{2} \)
89 \( 1 - 0.577T + 89T^{2} \)
97 \( 1 + 3.91T + 97T^{2} \)
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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

−9.228683244034658195551924562858, −8.116929899811195018990266400316, −7.70153515466961312643525646465, −6.40239919969037187005062020485, −5.90324039932996144543463661479, −5.15819517075313412821505476612, −4.17599570516765756930959501161, −3.58770445103657965574502239626, −2.34241904051139675505698096420, −1.91048050542531803716997485783, 1.91048050542531803716997485783, 2.34241904051139675505698096420, 3.58770445103657965574502239626, 4.17599570516765756930959501161, 5.15819517075313412821505476612, 5.90324039932996144543463661479, 6.40239919969037187005062020485, 7.70153515466961312643525646465, 8.116929899811195018990266400316, 9.228683244034658195551924562858

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