Properties

Label 2-1859-1.1-c1-0-53
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  − 0.603·2-s − 0.154·3-s − 1.63·4-s + 4.17·5-s + 0.0932·6-s + 2.73·7-s + 2.19·8-s − 2.97·9-s − 2.51·10-s + 11-s + 0.252·12-s − 1.65·14-s − 0.645·15-s + 1.94·16-s − 2.78·17-s + 1.79·18-s + 8.62·19-s − 6.83·20-s − 0.423·21-s − 0.603·22-s − 5.00·23-s − 0.338·24-s + 12.4·25-s + 0.923·27-s − 4.48·28-s + 5.70·29-s + 0.389·30-s + ⋯
L(s)  = 1  − 0.426·2-s − 0.0892·3-s − 0.818·4-s + 1.86·5-s + 0.0380·6-s + 1.03·7-s + 0.775·8-s − 0.992·9-s − 0.796·10-s + 0.301·11-s + 0.0729·12-s − 0.441·14-s − 0.166·15-s + 0.487·16-s − 0.676·17-s + 0.423·18-s + 1.97·19-s − 1.52·20-s − 0.0923·21-s − 0.128·22-s − 1.04·23-s − 0.0691·24-s + 2.48·25-s + 0.177·27-s − 0.847·28-s + 1.05·29-s + 0.0710·30-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

\(L(1)\) \(\approx\) \(1.778871171\)
\(L(\frac12)\) \(\approx\) \(1.778871171\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.603T + 2T^{2} \)
3 \( 1 + 0.154T + 3T^{2} \)
5 \( 1 - 4.17T + 5T^{2} \)
7 \( 1 - 2.73T + 7T^{2} \)
17 \( 1 + 2.78T + 17T^{2} \)
19 \( 1 - 8.62T + 19T^{2} \)
23 \( 1 + 5.00T + 23T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 + 3.15T + 37T^{2} \)
41 \( 1 - 1.15T + 41T^{2} \)
43 \( 1 + 6.42T + 43T^{2} \)
47 \( 1 + 0.354T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 9.99T + 61T^{2} \)
67 \( 1 - 8.44T + 67T^{2} \)
71 \( 1 - 5.99T + 71T^{2} \)
73 \( 1 + 4.94T + 73T^{2} \)
79 \( 1 + 7.84T + 79T^{2} \)
83 \( 1 - 2.14T + 83T^{2} \)
89 \( 1 - 6.72T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−9.351572875605977727119933034860, −8.508397298135571769748385050847, −8.042371565190255061582981206963, −6.76557787990542458277437127908, −5.88917315005032874348923095286, −5.19971286199733425466784230653, −4.67122193215628608733228533171, −3.13379495702853415176306315620, −1.98581721617233015786765043763, −1.06125610377382545421221103059, 1.06125610377382545421221103059, 1.98581721617233015786765043763, 3.13379495702853415176306315620, 4.67122193215628608733228533171, 5.19971286199733425466784230653, 5.88917315005032874348923095286, 6.76557787990542458277437127908, 8.042371565190255061582981206963, 8.508397298135571769748385050847, 9.351572875605977727119933034860

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