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Properties

Label 2-702-117.25-c1-0-0
Degree $2$
Conductor $702$
Sign $-0.825 + 0.564i$
Analytic cond. $5.60549$
Root an. cond. $2.36759$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.53 + 1.46i)5-s + (−1.90 − 1.10i)7-s + 0.999i·8-s − 2.93·10-s + (−4.47 − 2.58i)11-s + (2.72 + 2.36i)13-s + (−1.10 − 1.90i)14-s + (−0.5 + 0.866i)16-s − 2.31·17-s − 5.16i·19-s + (−2.53 − 1.46i)20-s + (−2.58 − 4.47i)22-s + (−4.19 − 7.26i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.13 + 0.655i)5-s + (−0.721 − 0.416i)7-s + 0.353i·8-s − 0.927·10-s + (−1.34 − 0.779i)11-s + (0.756 + 0.654i)13-s + (−0.294 − 0.510i)14-s + (−0.125 + 0.216i)16-s − 0.560·17-s − 1.18i·19-s + (−0.567 − 0.327i)20-s + (−0.551 − 0.954i)22-s + (−0.874 − 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.825 + 0.564i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0216818 - 0.0701199i\)
\(L(\frac12)\) \(\approx\) \(0.0216818 - 0.0701199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
13 \( 1 + (-2.72 - 2.36i)T \)
good5 \( 1 + (2.53 - 1.46i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.90 + 1.10i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.47 + 2.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.31T + 17T^{2} \)
19 \( 1 + 5.16iT - 19T^{2} \)
23 \( 1 + (4.19 + 7.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.38 - 3.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.646iT - 37T^{2} \)
41 \( 1 + (-0.674 + 0.389i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.74 - 3.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.79 - 2.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.68T + 53T^{2} \)
59 \( 1 + (2.59 - 1.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.432 + 0.748i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.68 - 5.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 4.27iT - 73T^{2} \)
79 \( 1 + (-1.52 + 2.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.19 - 1.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.34iT - 89T^{2} \)
97 \( 1 + (1.29 + 0.745i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.91418090689632768330571933177, −10.54417325383799896646913450187, −8.971939805257857309466965147036, −8.242137822398209029461952453072, −7.19116766344571294311318392679, −6.74892368832982868824557717129, −5.62913607414179407495735212081, −4.40445350924271334878143405675, −3.55180152485905182895507933821, −2.69337460696547808001679594947, 0.02914097709753215586055191980, 2.08342811135212316288897829218, 3.48568594458314646862656594380, 4.15427622228232080913573857641, 5.38275675890465053612247459876, 6.01998086882489578203638402056, 7.58700410257097133288703291513, 7.937257367959234912308395949455, 9.181112209005517228147033682774, 10.05166045889698800720693232194

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