Properties

Label 2-1859-1.1-c1-0-28
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  − 1.97·2-s − 0.571·3-s + 1.89·4-s − 1.23·5-s + 1.12·6-s + 2.55·7-s + 0.199·8-s − 2.67·9-s + 2.42·10-s + 11-s − 1.08·12-s − 5.05·14-s + 0.703·15-s − 4.19·16-s + 5.97·17-s + 5.27·18-s + 5.32·19-s − 2.33·20-s − 1.46·21-s − 1.97·22-s − 3.26·23-s − 0.113·24-s − 3.48·25-s + 3.24·27-s + 4.85·28-s − 2.86·29-s − 1.38·30-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.329·3-s + 0.949·4-s − 0.550·5-s + 0.460·6-s + 0.966·7-s + 0.0704·8-s − 0.891·9-s + 0.768·10-s + 0.301·11-s − 0.313·12-s − 1.34·14-s + 0.181·15-s − 1.04·16-s + 1.44·17-s + 1.24·18-s + 1.22·19-s − 0.522·20-s − 0.319·21-s − 0.420·22-s − 0.680·23-s − 0.0232·24-s − 0.697·25-s + 0.623·27-s + 0.918·28-s − 0.531·29-s − 0.253·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6558410306\)
\(L(\frac12)\) \(\approx\) \(0.6558410306\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.97T + 2T^{2} \)
3 \( 1 + 0.571T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + 2.90T + 31T^{2} \)
37 \( 1 - 9.62T + 37T^{2} \)
41 \( 1 - 3.47T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 0.858T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 8.76T + 59T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 + 3.64T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 5.63T + 73T^{2} \)
79 \( 1 - 7.57T + 79T^{2} \)
83 \( 1 - 0.707T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 3.45T + 97T^{2} \)
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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

−9.407194918673290464839797550726, −8.222786460206997006740969472733, −7.911069774962083041087146874011, −7.38527505002233823127076118184, −6.10756088533602689363667298770, −5.33660975450829493838282839091, −4.32039115993984026991395554841, −3.14869898770133128956153333974, −1.76500988075011739284750518644, −0.71288728839891128672135862686, 0.71288728839891128672135862686, 1.76500988075011739284750518644, 3.14869898770133128956153333974, 4.32039115993984026991395554841, 5.33660975450829493838282839091, 6.10756088533602689363667298770, 7.38527505002233823127076118184, 7.911069774962083041087146874011, 8.222786460206997006740969472733, 9.407194918673290464839797550726

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