Properties

Label 2-1859-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.26·3-s − 4-s − 2.11·5-s + 2.26·6-s − 3.37·7-s + 3·8-s + 2.11·9-s + 2.11·10-s − 11-s + 2.26·12-s + 3.37·14-s + 4.78·15-s − 16-s − 2.14·17-s − 2.11·18-s − 3.14·19-s + 2.11·20-s + 7.63·21-s + 22-s + 4.52·23-s − 6.78·24-s − 0.523·25-s + 2·27-s + 3.37·28-s + 3.49·29-s − 4.78·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.30·3-s − 0.5·4-s − 0.946·5-s + 0.923·6-s − 1.27·7-s + 1.06·8-s + 0.705·9-s + 0.669·10-s − 0.301·11-s + 0.652·12-s + 0.902·14-s + 1.23·15-s − 0.250·16-s − 0.520·17-s − 0.498·18-s − 0.721·19-s + 0.473·20-s + 1.66·21-s + 0.213·22-s + 0.943·23-s − 1.38·24-s − 0.104·25-s + 0.384·27-s + 0.638·28-s + 0.648·29-s − 0.873·30-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: \(1\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + 2.26T + 3T^{2} \)
5 \( 1 + 2.11T + 5T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + 3.14T + 19T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 - 9.27T + 31T^{2} \)
37 \( 1 - 9.16T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 - 1.96T + 47T^{2} \)
53 \( 1 - 7.75T + 53T^{2} \)
59 \( 1 - 6.03T + 59T^{2} \)
61 \( 1 - 9.78T + 61T^{2} \)
67 \( 1 + 4.03T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 1.66T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 + 9.34T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.763076018552346604484334740361, −8.237362503111811048225216525344, −7.14964681070961041311026470575, −6.59243559512965596320987937104, −5.70109373622720831622957883267, −4.68448815619780475066717924515, −4.08729924007336391120061870774, −2.83659294524702960312553979956, −0.837200078331154618820739186820, 0, 0.837200078331154618820739186820, 2.83659294524702960312553979956, 4.08729924007336391120061870774, 4.68448815619780475066717924515, 5.70109373622720831622957883267, 6.59243559512965596320987937104, 7.14964681070961041311026470575, 8.237362503111811048225216525344, 8.763076018552346604484334740361

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