Properties

Label 2-1859-1.1-c1-0-26
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  + 2.14·2-s − 1.72·3-s + 2.59·4-s − 3.54·5-s − 3.69·6-s − 2.57·7-s + 1.28·8-s − 0.0251·9-s − 7.60·10-s + 11-s − 4.48·12-s − 5.51·14-s + 6.11·15-s − 2.44·16-s + 4.72·17-s − 0.0539·18-s + 4.99·19-s − 9.21·20-s + 4.43·21-s + 2.14·22-s + 4.05·23-s − 2.21·24-s + 7.57·25-s + 5.21·27-s − 6.67·28-s + 7.00·29-s + 13.1·30-s + ⋯
L(s)  = 1  + 1.51·2-s − 0.995·3-s + 1.29·4-s − 1.58·5-s − 1.50·6-s − 0.971·7-s + 0.453·8-s − 0.00838·9-s − 2.40·10-s + 0.301·11-s − 1.29·12-s − 1.47·14-s + 1.57·15-s − 0.611·16-s + 1.14·17-s − 0.0127·18-s + 1.14·19-s − 2.06·20-s + 0.967·21-s + 0.457·22-s + 0.846·23-s − 0.451·24-s + 1.51·25-s + 1.00·27-s − 1.26·28-s + 1.30·29-s + 2.39·30-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

\(L(1)\) \(\approx\) \(1.558843825\)
\(L(\frac12)\) \(\approx\) \(1.558843825\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.14T + 2T^{2} \)
3 \( 1 + 1.72T + 3T^{2} \)
5 \( 1 + 3.54T + 5T^{2} \)
7 \( 1 + 2.57T + 7T^{2} \)
17 \( 1 - 4.72T + 17T^{2} \)
19 \( 1 - 4.99T + 19T^{2} \)
23 \( 1 - 4.05T + 23T^{2} \)
29 \( 1 - 7.00T + 29T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 - 1.93T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 4.48T + 53T^{2} \)
59 \( 1 - 9.21T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 4.22T + 71T^{2} \)
73 \( 1 + 0.634T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 - 2.96T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−9.263596050224324342940023255084, −8.269966675263442672910656057992, −7.14573470876276277360438314938, −6.78578324829622767332740744224, −5.73998044682098573444374744675, −5.21135993249101630320426049819, −4.32419412084699179430523645595, −3.42301670179029303153833476495, −3.04451087452385196976670946502, −0.69205309741174080130631162713, 0.69205309741174080130631162713, 3.04451087452385196976670946502, 3.42301670179029303153833476495, 4.32419412084699179430523645595, 5.21135993249101630320426049819, 5.73998044682098573444374744675, 6.78578324829622767332740744224, 7.14573470876276277360438314938, 8.269966675263442672910656057992, 9.263596050224324342940023255084

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