↑

Properties

Label 2-714-17.16-c1-0-3
Degree $2$
Conductor $714$
Sign $-0.817 - 0.575i$
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  − 2-s + i·3-s + 4-s + 4.30i·5-s − i·6-s − i·7-s − 8-s − 9-s − 4.30i·10-s + 3.17i·11-s + i·12-s + 5.03·13-s + i·14-s − 4.30·15-s + 16-s + (3.37 + 2.37i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.92i·5-s − 0.408i·6-s − 0.377i·7-s − 0.353·8-s − 0.333·9-s − 1.36i·10-s + 0.956i·11-s + 0.288i·12-s + 1.39·13-s + 0.267i·14-s − 1.11·15-s + 0.250·16-s + (0.817 + 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $-0.817 - 0.575i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ -0.817 - 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305330 + 0.964712i\)
\(L(\frac12)\) \(\approx\) \(0.305330 + 0.964712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
7 \( 1 + iT \)
17 \( 1 + (-3.37 - 2.37i)T \)
good5 \( 1 - 4.30iT - 5T^{2} \)
11 \( 1 - 3.17iT - 11T^{2} \)
13 \( 1 - 5.03T + 13T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 4.74iT - 23T^{2} \)
29 \( 1 + 7.33iT - 29T^{2} \)
31 \( 1 - 8.60iT - 31T^{2} \)
37 \( 1 + 7.03iT - 37T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + 0.442T + 43T^{2} \)
47 \( 1 + 8.60T + 47T^{2} \)
53 \( 1 + 6.44T + 53T^{2} \)
59 \( 1 - 3.47T + 59T^{2} \)
61 \( 1 - 6.20iT - 61T^{2} \)
67 \( 1 + 0.442T + 67T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + 5.76iT - 73T^{2} \)
79 \( 1 + 4.16iT - 79T^{2} \)
83 \( 1 - 0.686T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 2.84iT - 97T^{2} \)
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

−10.51603783392826054264498496267, −10.18535847058422500742204815681, −9.276058529890220535402445136532, −8.080752871740193339999200272809, −7.32010551392191636109504954159, −6.53102185849596434993338966433, −5.70591816679464434559695745291, −3.92147203850355008552287793268, −3.27002763565053171097429686442, −1.92322785020501700643898345916, 0.70229918025259791665305719740, 1.58901123984003538650681521066, 3.27731310724334535165509939295, 4.74524391185678103265429534926, 5.74575641351253242323772265701, 6.43976197595746796871194988864, 8.066917007534365969928129815236, 8.266565827369839463109667977111, 9.014346149714385380382194132041, 9.755962769154296998026291692473

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