Properties

Label 2-1859-1.1-c1-0-80
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  + 2.68·2-s − 2.90·3-s + 5.22·4-s + 3.09·5-s − 7.82·6-s + 1.70·7-s + 8.66·8-s + 5.46·9-s + 8.31·10-s + 11-s − 15.2·12-s + 4.58·14-s − 9.00·15-s + 12.8·16-s + 1.04·17-s + 14.6·18-s − 0.291·19-s + 16.1·20-s − 4.95·21-s + 2.68·22-s − 7.88·23-s − 25.2·24-s + 4.57·25-s − 7.17·27-s + 8.90·28-s − 0.935·29-s − 24.1·30-s + ⋯
L(s)  = 1  + 1.90·2-s − 1.68·3-s + 2.61·4-s + 1.38·5-s − 3.19·6-s + 0.644·7-s + 3.06·8-s + 1.82·9-s + 2.62·10-s + 0.301·11-s − 4.38·12-s + 1.22·14-s − 2.32·15-s + 3.21·16-s + 0.252·17-s + 3.46·18-s − 0.0667·19-s + 3.61·20-s − 1.08·21-s + 0.573·22-s − 1.64·23-s − 5.14·24-s + 0.914·25-s − 1.38·27-s + 1.68·28-s − 0.173·29-s − 4.41·30-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

\(L(1)\) \(\approx\) \(4.843255772\)
\(L(\frac12)\) \(\approx\) \(4.843255772\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 2T^{2} \)
3 \( 1 + 2.90T + 3T^{2} \)
5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 - 1.70T + 7T^{2} \)
17 \( 1 - 1.04T + 17T^{2} \)
19 \( 1 + 0.291T + 19T^{2} \)
23 \( 1 + 7.88T + 23T^{2} \)
29 \( 1 + 0.935T + 29T^{2} \)
31 \( 1 - 1.01T + 31T^{2} \)
37 \( 1 + 6.79T + 37T^{2} \)
41 \( 1 - 4.76T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 2.04T + 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 9.48T + 61T^{2} \)
67 \( 1 - 9.43T + 67T^{2} \)
71 \( 1 - 0.582T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 5.79T + 89T^{2} \)
97 \( 1 + 0.922T + 97T^{2} \)
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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

−9.827592375266782604554463777181, −8.121508448975128849023689160899, −6.88764644847844978180949146412, −6.49125098078374360340142930801, −5.64426390390831888795870701639, −5.42272911170412415053760104483, −4.63227471385997061686226447431, −3.75233532098444110936882463897, −2.22389768627036907976256747365, −1.47267932417951160364159962718, 1.47267932417951160364159962718, 2.22389768627036907976256747365, 3.75233532098444110936882463897, 4.63227471385997061686226447431, 5.42272911170412415053760104483, 5.64426390390831888795870701639, 6.49125098078374360340142930801, 6.88764644847844978180949146412, 8.121508448975128849023689160899, 9.827592375266782604554463777181

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