↑

Properties

Label 2-714-17.13-c1-0-7
Degree $2$
Conductor $714$
Sign $0.367 - 0.930i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.943 + 0.943i)5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s − i·8-s + 1.00i·9-s + (−0.943 + 0.943i)10-s + (−1.94 + 1.94i)11-s + (0.707 + 0.707i)12-s + 5.16·13-s + (0.707 + 0.707i)14-s − 1.33i·15-s + 16-s + (−1.16 − 3.95i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.421 + 0.421i)5-s + (0.288 − 0.288i)6-s + (0.267 − 0.267i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.298 + 0.298i)10-s + (−0.585 + 0.585i)11-s + (0.204 + 0.204i)12-s + 1.43·13-s + (0.188 + 0.188i)14-s − 0.344i·15-s + 0.250·16-s + (−0.282 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14287 + 0.777251i\)
\(L(\frac12)\) \(\approx\) \(1.14287 + 0.777251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (1.16 + 3.95i)T \)
good5 \( 1 + (-0.943 - 0.943i)T + 5iT^{2} \)
11 \( 1 + (1.94 - 1.94i)T - 11iT^{2} \)
13 \( 1 - 5.16T + 13T^{2} \)
19 \( 1 - 5.94iT - 19T^{2} \)
23 \( 1 + (-1.04 + 1.04i)T - 23iT^{2} \)
29 \( 1 + (-5.04 - 5.04i)T + 29iT^{2} \)
31 \( 1 + (-6.63 - 6.63i)T + 31iT^{2} \)
37 \( 1 + (-5.06 - 5.06i)T + 37iT^{2} \)
41 \( 1 + (1.84 - 1.84i)T - 41iT^{2} \)
43 \( 1 - 4.84iT - 43T^{2} \)
47 \( 1 + 4.35T + 47T^{2} \)
53 \( 1 + 8.34iT - 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 + (-5.43 + 5.43i)T - 61iT^{2} \)
67 \( 1 + 0.646T + 67T^{2} \)
71 \( 1 + (-4.92 - 4.92i)T + 71iT^{2} \)
73 \( 1 + (1.02 + 1.02i)T + 73iT^{2} \)
79 \( 1 + (-5.04 + 5.04i)T - 79iT^{2} \)
83 \( 1 - 6.49iT - 83T^{2} \)
89 \( 1 + 4.94T + 89T^{2} \)
97 \( 1 + (12.4 + 12.4i)T + 97iT^{2} \)
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

−10.42911088134347547432556800838, −9.891841156942160082119153612579, −8.524860646248397751193783523273, −7.994436514096310292018449440107, −6.82215835486708244704571623846, −6.39309280837298812752244461370, −5.33446657957497857853065276724, −4.43256851496600758645287739439, −2.95379803198189842803566425553, −1.31239220223900765985344025818, 0.916647377952152762149870564177, 2.42715325810538820821696606317, 3.72096684597719027417296981333, 4.71118379085761618306452555142, 5.66246531434053381063487934448, 6.38145205061114279960051225253, 8.021743860359860409737337739051, 8.736754532127593906579094187866, 9.454944015506203973124989858322, 10.43926613401033984977901289389

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