↑

Properties

Label 2-1710-19.11-c1-0-16
Degree $2$
Conductor $1710$
Sign $0.305 - 0.952i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 4.35·7-s + 0.999·8-s + (−0.499 − 0.866i)10-s + 3·11-s + (2 + 3.46i)13-s + (−2.17 + 3.77i)14-s + (−0.5 + 0.866i)16-s + (−2.67 + 4.64i)17-s + (2.17 − 3.77i)19-s + 0.999·20-s + (−1.5 + 2.59i)22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 1.64·7-s + 0.353·8-s + (−0.158 − 0.273i)10-s + 0.904·11-s + (0.554 + 0.960i)13-s + (−0.582 + 1.00i)14-s + (−0.125 + 0.216i)16-s + (−0.649 + 1.12i)17-s + (0.499 − 0.866i)19-s + 0.223·20-s + (−0.319 + 0.553i)22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.305 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765643910\)
\(L(\frac12)\) \(\approx\) \(1.765643910\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-2.17 + 3.77i)T \)
good7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.67 - 4.64i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.320 - 0.555i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.71T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-1.17 + 2.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.53 - 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.35 - 4.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.67 - 2.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.32 + 2.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.03 + 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.67 - 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (4.17 + 7.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.03 - 10.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.070574425831993915836573513231, −8.704914275778696526072856361225, −7.987596220989705549285617068884, −7.05464046633106616783150922717, −6.51121047433674012319150545233, −5.50874423631280044953100603671, −4.47824211758098739271702233254, −3.96196281138290939690853168747, −2.20413240355772619200188678044, −1.22812109928839846335651535017, 0.949318353104839702202344454608, 1.77593479206701776949079485089, 3.10535860185766406535545047873, 4.17277164585643775242739971941, 4.87789492362007687546880054117, 5.75125018720201358222613698630, 7.01041179218394589039752669008, 8.018177900489844467644323214783, 8.251723984172333350202806521940, 9.172850994149029756091302491425

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