↑

Properties

Label 2-2695-2695.2584-c0-0-3
Degree $2$
Conductor $2695$
Sign $0.462 - 0.886i$
Analytic cond. $1.34498$
Root an. cond. $1.15973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.193 + 0.846i)2-s + (0.222 − 0.107i)4-s + (−0.623 − 0.781i)5-s + (0.974 + 0.222i)7-s + (0.674 + 0.846i)8-s + (−0.222 + 0.974i)9-s + (0.541 − 0.678i)10-s + (0.222 + 0.974i)11-s + (−0.433 − 1.90i)13-s + 0.867i·14-s + (−0.431 + 0.541i)16-s + (1.40 + 0.678i)17-s − 0.867·18-s + (−0.222 − 0.107i)20-s + (−0.781 + 0.376i)22-s + ⋯
L(s)  = 1  + (0.193 + 0.846i)2-s + (0.222 − 0.107i)4-s + (−0.623 − 0.781i)5-s + (0.974 + 0.222i)7-s + (0.674 + 0.846i)8-s + (−0.222 + 0.974i)9-s + (0.541 − 0.678i)10-s + (0.222 + 0.974i)11-s + (−0.433 − 1.90i)13-s + 0.867i·14-s + (−0.431 + 0.541i)16-s + (1.40 + 0.678i)17-s − 0.867·18-s + (−0.222 − 0.107i)20-s + (−0.781 + 0.376i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $0.462 - 0.886i$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2695} (2584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.605805098\)
\(L(\frac12)\) \(\approx\) \(1.605805098\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.974 - 0.222i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
good2 \( 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2} \)
3 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-1.21 + 1.52i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.347 - 1.52i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
97 \( 1 - T^{2} \)
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

−8.763545937757817608504459275832, −8.063460461137399024774372235727, −7.62897326907777891151978434935, −7.28165533929102627828688190310, −5.62910206702146335522453625094, −5.45642088636639488650001204696, −4.83326143791814100884173319461, −3.78699706385628723020671909674, −2.38180954949052421707082172288, −1.42272115654141410226558327340, 1.15297601978162516498118864847, 2.26872439675699122113991690909, 3.31455803751174280386847312759, 3.80533413011371039449391189102, 4.62472237486082719501214007399, 5.88269617058099557546527992106, 6.77332971620844935882865908826, 7.34079131916835929851508694474, 7.994011890078519224331911732448, 9.112109645667362200632082467548

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