Properties

Label 2-1859-1.1-c1-0-15
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  − 1.53·2-s − 3.28·3-s + 0.361·4-s − 1.21·5-s + 5.04·6-s − 1.98·7-s + 2.51·8-s + 7.77·9-s + 1.86·10-s + 11-s − 1.18·12-s + 3.05·14-s + 3.97·15-s − 4.59·16-s + 6.12·17-s − 11.9·18-s + 3.70·19-s − 0.438·20-s + 6.51·21-s − 1.53·22-s − 5.04·23-s − 8.26·24-s − 3.52·25-s − 15.6·27-s − 0.718·28-s + 8.47·29-s − 6.11·30-s + ⋯
L(s)  = 1  − 1.08·2-s − 1.89·3-s + 0.180·4-s − 0.542·5-s + 2.05·6-s − 0.750·7-s + 0.890·8-s + 2.59·9-s + 0.589·10-s + 0.301·11-s − 0.342·12-s + 0.815·14-s + 1.02·15-s − 1.14·16-s + 1.48·17-s − 2.81·18-s + 0.849·19-s − 0.0980·20-s + 1.42·21-s − 0.327·22-s − 1.05·23-s − 1.68·24-s − 0.705·25-s − 3.01·27-s − 0.135·28-s + 1.57·29-s − 1.11·30-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

\(L(1)\) \(\approx\) \(0.2825813241\)
\(L(\frac12)\) \(\approx\) \(0.2825813241\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 2T^{2} \)
3 \( 1 + 3.28T + 3T^{2} \)
5 \( 1 + 1.21T + 5T^{2} \)
7 \( 1 + 1.98T + 7T^{2} \)
17 \( 1 - 6.12T + 17T^{2} \)
19 \( 1 - 3.70T + 19T^{2} \)
23 \( 1 + 5.04T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 + 8.15T + 37T^{2} \)
41 \( 1 + 3.80T + 41T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 - 0.832T + 67T^{2} \)
71 \( 1 + 5.22T + 71T^{2} \)
73 \( 1 - 9.29T + 73T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−9.670765891478217759252555073390, −8.386363240028691915420950616474, −7.60776564886085670276393832977, −6.92392196390958626840952389275, −6.15114678437803632647187946814, −5.32283261187403004245688997637, −4.47544179032467087801181518337, −3.51935750704008040419147403337, −1.45216500703899538462153790839, −0.51232493470001687715764340267, 0.51232493470001687715764340267, 1.45216500703899538462153790839, 3.51935750704008040419147403337, 4.47544179032467087801181518337, 5.32283261187403004245688997637, 6.15114678437803632647187946814, 6.92392196390958626840952389275, 7.60776564886085670276393832977, 8.386363240028691915420950616474, 9.670765891478217759252555073390

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