Properties

Label 2-1859-1.1-c1-0-30
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s + 0.960·3-s + 5.32·4-s − 3.57·5-s − 2.59·6-s + 3.95·7-s − 8.99·8-s − 2.07·9-s + 9.66·10-s + 11-s + 5.11·12-s − 10.7·14-s − 3.43·15-s + 13.6·16-s + 2.79·17-s + 5.62·18-s − 0.00685·19-s − 19.0·20-s + 3.79·21-s − 2.70·22-s − 3.94·23-s − 8.63·24-s + 7.76·25-s − 4.87·27-s + 21.0·28-s + 6.09·29-s + 9.28·30-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.554·3-s + 2.66·4-s − 1.59·5-s − 1.06·6-s + 1.49·7-s − 3.17·8-s − 0.692·9-s + 3.05·10-s + 0.301·11-s + 1.47·12-s − 2.86·14-s − 0.885·15-s + 3.42·16-s + 0.678·17-s + 1.32·18-s − 0.00157·19-s − 4.25·20-s + 0.828·21-s − 0.576·22-s − 0.822·23-s − 1.76·24-s + 1.55·25-s − 0.938·27-s + 3.97·28-s + 1.13·29-s + 1.69·30-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

\(L(1)\) \(\approx\) \(0.6627584728\)
\(L(\frac12)\) \(\approx\) \(0.6627584728\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 - 0.960T + 3T^{2} \)
5 \( 1 + 3.57T + 5T^{2} \)
7 \( 1 - 3.95T + 7T^{2} \)
17 \( 1 - 2.79T + 17T^{2} \)
19 \( 1 + 0.00685T + 19T^{2} \)
23 \( 1 + 3.94T + 23T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
31 \( 1 - 0.242T + 31T^{2} \)
37 \( 1 - 0.801T + 37T^{2} \)
41 \( 1 + 6.58T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 - 6.29T + 47T^{2} \)
53 \( 1 - 9.02T + 53T^{2} \)
59 \( 1 - 4.00T + 59T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 - 8.97T + 67T^{2} \)
71 \( 1 - 8.03T + 71T^{2} \)
73 \( 1 - 0.166T + 73T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 - 0.547T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 8.78T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.786770801103578616579977723900, −8.449079650592316530558641053592, −7.925642233383616294443127180197, −7.51217453131576372278380881396, −6.56518750443513373311422714874, −5.29926624134864001405655422373, −3.98803729681005337270279840798, −2.99608103748976102430838532014, −1.90283266394824333722742977292, −0.71470258001110782496311380970, 0.71470258001110782496311380970, 1.90283266394824333722742977292, 2.99608103748976102430838532014, 3.98803729681005337270279840798, 5.29926624134864001405655422373, 6.56518750443513373311422714874, 7.51217453131576372278380881396, 7.925642233383616294443127180197, 8.449079650592316530558641053592, 8.786770801103578616579977723900

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