Properties

Label 2-1859-1.1-c1-0-5
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  + 0.149·2-s − 3.21·3-s − 1.97·4-s + 1.07·5-s − 0.479·6-s − 3.58·7-s − 0.593·8-s + 7.36·9-s + 0.160·10-s + 11-s + 6.36·12-s − 0.534·14-s − 3.46·15-s + 3.86·16-s − 5.78·17-s + 1.09·18-s − 7.32·19-s − 2.12·20-s + 11.5·21-s + 0.149·22-s − 5.23·23-s + 1.90·24-s − 3.84·25-s − 14.0·27-s + 7.08·28-s − 3.62·29-s − 0.516·30-s + ⋯
L(s)  = 1  + 0.105·2-s − 1.85·3-s − 0.988·4-s + 0.481·5-s − 0.195·6-s − 1.35·7-s − 0.209·8-s + 2.45·9-s + 0.0507·10-s + 0.301·11-s + 1.83·12-s − 0.142·14-s − 0.894·15-s + 0.966·16-s − 1.40·17-s + 0.258·18-s − 1.68·19-s − 0.475·20-s + 2.51·21-s + 0.0317·22-s − 1.09·23-s + 0.389·24-s − 0.768·25-s − 2.70·27-s + 1.33·28-s − 0.673·29-s − 0.0942·30-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

\(L(1)\) \(\approx\) \(0.1825392501\)
\(L(\frac12)\) \(\approx\) \(0.1825392501\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.149T + 2T^{2} \)
3 \( 1 + 3.21T + 3T^{2} \)
5 \( 1 - 1.07T + 5T^{2} \)
7 \( 1 + 3.58T + 7T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 + 7.32T + 19T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + 3.62T + 29T^{2} \)
31 \( 1 + 2.65T + 31T^{2} \)
37 \( 1 + 3.91T + 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 1.21T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 - 0.145T + 53T^{2} \)
59 \( 1 + 2.35T + 59T^{2} \)
61 \( 1 - 9.46T + 61T^{2} \)
67 \( 1 + 5.14T + 67T^{2} \)
71 \( 1 + 0.0702T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 + 3.82T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−9.380532215004770874427548122608, −8.752519495569772926127322960083, −7.34978606300870004730277960763, −6.37538881882222827142408888685, −6.14210670081655965146391385184, −5.35467192293402362481197082177, −4.32044256592303216070749664441, −3.86102902052414884581489944486, −1.98696540121566704007326017880, −0.30152771366293336387669340468, 0.30152771366293336387669340468, 1.98696540121566704007326017880, 3.86102902052414884581489944486, 4.32044256592303216070749664441, 5.35467192293402362481197082177, 6.14210670081655965146391385184, 6.37538881882222827142408888685, 7.34978606300870004730277960763, 8.752519495569772926127322960083, 9.380532215004770874427548122608

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