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Properties

Label 2-702-117.25-c1-0-4
Degree $2$
Conductor $702$
Sign $0.428 + 0.903i$
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 + (−3.32 + 1.91i)5-s + (−2.91 − 1.68i)7-s − 0.999i·8-s + 3.83·10-s + (1.72 + 0.995i)11-s + (3.49 + 0.880i)13-s + (1.68 + 2.91i)14-s + (−0.5 + 0.866i)16-s − 2.60·17-s + 1.99i·19-s + (−3.32 − 1.91i)20-s + (−0.995 − 1.72i)22-s + (−2.13 − 3.70i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.48 + 0.857i)5-s + (−1.10 − 0.636i)7-s − 0.353i·8-s + 1.21·10-s + (0.520 + 0.300i)11-s + (0.969 + 0.244i)13-s + (0.450 + 0.779i)14-s + (−0.125 + 0.216i)16-s − 0.630·17-s + 0.456i·19-s + (−0.742 − 0.428i)20-s + (−0.212 − 0.367i)22-s + (−0.445 − 0.771i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $0.428 + 0.903i$
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: \((2,\ 702,\ (\ :1/2),\ 0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495258 - 0.313065i\)
\(L(\frac12)\) \(\approx\) \(0.495258 - 0.313065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.49 - 0.880i)T \)
good5 \( 1 + (3.32 - 1.91i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.91 + 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.72 - 0.995i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 - 1.99iT - 19T^{2} \)
23 \( 1 + (2.13 + 3.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.57 + 2.64i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.08iT - 37T^{2} \)
41 \( 1 + (-7.13 + 4.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.13 - 8.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.22 + 2.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.16T + 53T^{2} \)
59 \( 1 + (-6.33 + 3.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.63 + 9.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.90 + 2.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 - 8.41iT - 73T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.03 - 1.17i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.28iT - 89T^{2} \)
97 \( 1 + (8.92 + 5.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31636938520224608085741866838, −9.613425832031931409370214604741, −8.434280507210672496979005709212, −7.81722770400260089854390887252, −6.71350712447344521037839981287, −6.44655752576476273821290812616, −4.10581842945635946969802896543, −3.81491816730474313028911601533, −2.60024894810012612308932049761, −0.49882110520650159605069433620, 0.993447253172592553352107689984, 3.08711067028120792350937418376, 4.06302203104362554518051056346, 5.26776714119381307785808733379, 6.37663552038364316783846739718, 7.12533622091048767586662920348, 8.366907529585688495123587412324, 8.645720427369149130832734426273, 9.437324350144910190672188566394, 10.55945926782172085622374705941

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