Properties

Label 2-1859-1.1-c1-0-44
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  − 2.32·2-s + 2.00·3-s + 3.40·4-s + 3.04·5-s − 4.65·6-s − 5.08·7-s − 3.26·8-s + 1.00·9-s − 7.08·10-s + 11-s + 6.81·12-s + 11.8·14-s + 6.09·15-s + 0.781·16-s + 4.98·17-s − 2.33·18-s − 4.25·19-s + 10.3·20-s − 10.1·21-s − 2.32·22-s + 3.33·23-s − 6.53·24-s + 4.28·25-s − 3.99·27-s − 17.3·28-s + 0.0575·29-s − 14.1·30-s + ⋯
L(s)  = 1  − 1.64·2-s + 1.15·3-s + 1.70·4-s + 1.36·5-s − 1.89·6-s − 1.92·7-s − 1.15·8-s + 0.334·9-s − 2.24·10-s + 0.301·11-s + 1.96·12-s + 3.16·14-s + 1.57·15-s + 0.195·16-s + 1.20·17-s − 0.550·18-s − 0.976·19-s + 2.32·20-s − 2.22·21-s − 0.495·22-s + 0.696·23-s − 1.33·24-s + 0.857·25-s − 0.768·27-s − 3.27·28-s + 0.0106·29-s − 2.58·30-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

\(L(1)\) \(\approx\) \(1.223728405\)
\(L(\frac12)\) \(\approx\) \(1.223728405\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.32T + 2T^{2} \)
3 \( 1 - 2.00T + 3T^{2} \)
5 \( 1 - 3.04T + 5T^{2} \)
7 \( 1 + 5.08T + 7T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 + 4.25T + 19T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 - 0.0575T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 0.344T + 43T^{2} \)
47 \( 1 - 5.15T + 47T^{2} \)
53 \( 1 - 5.27T + 53T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 0.194T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 9.71T + 73T^{2} \)
79 \( 1 + 3.22T + 79T^{2} \)
83 \( 1 - 7.54T + 83T^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 + 3.31T + 97T^{2} \)
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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

−9.300452314347684814566844973158, −8.793894217766409976560063684617, −7.950508349927862239017701455667, −7.01636065491250999243416674988, −6.38539614443282441812733384406, −5.67326640273837019848820283424, −3.80127958556579066820134048143, −2.75831156016787631291471612967, −2.28088928231353786887212719366, −0.903344307775768006244395466344, 0.903344307775768006244395466344, 2.28088928231353786887212719366, 2.75831156016787631291471612967, 3.80127958556579066820134048143, 5.67326640273837019848820283424, 6.38539614443282441812733384406, 7.01636065491250999243416674988, 7.950508349927862239017701455667, 8.793894217766409976560063684617, 9.300452314347684814566844973158

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