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Properties

Label 2-790-395.339-c1-0-0
Degree $2$
Conductor $790$
Sign $-0.164 + 0.986i$
Analytic cond. $6.30818$
Root an. cond. $2.51160$
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 + (−2.91 + 1.68i)3-s + (0.499 − 0.866i)4-s + (−1.75 − 1.38i)5-s + (1.68 − 2.91i)6-s + (3.40 + 1.96i)7-s + 0.999i·8-s + (4.18 − 7.24i)9-s + (2.21 + 0.322i)10-s + (−3.03 + 5.25i)11-s + 3.37i·12-s + (−1.99 + 1.14i)13-s − 3.92·14-s + (7.46 + 1.08i)15-s + (−0.5 − 0.866i)16-s − 1.60i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−1.68 + 0.973i)3-s + (0.249 − 0.433i)4-s + (−0.784 − 0.619i)5-s + (0.688 − 1.19i)6-s + (1.28 + 0.742i)7-s + 0.353i·8-s + (1.39 − 2.41i)9-s + (0.699 + 0.101i)10-s + (−0.915 + 1.58i)11-s + 0.973i·12-s + (−0.552 + 0.318i)13-s − 1.04·14-s + (1.92 + 0.280i)15-s + (−0.125 − 0.216i)16-s − 0.388i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(790\)    =    \(2 \cdot 5 \cdot 79\)
Sign: $-0.164 + 0.986i$
Analytic conductor: \(6.30818\)
Root analytic conductor: \(2.51160\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{790} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 790,\ (\ :1/2),\ -0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0337857 - 0.0398835i\)
\(L(\frac12)\) \(\approx\) \(0.0337857 - 0.0398835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.75 + 1.38i)T \)
79 \( 1 + (7.11 + 5.32i)T \)
good3 \( 1 + (2.91 - 1.68i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.40 - 1.96i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.03 - 5.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.99 - 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.60iT - 17T^{2} \)
19 \( 1 + (2.86 - 4.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.462 + 0.267i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.38 + 4.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.06 + 1.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.800T + 41T^{2} \)
43 \( 1 + (-0.198 + 0.114i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.93 + 5.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.08 + 0.629i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.14 + 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.78T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 7.98T + 71T^{2} \)
73 \( 1 + (-2.91 - 1.68i)T + (36.5 + 63.2i)T^{2} \)
83 \( 1 + (-3.26 - 1.88i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 8.67iT - 97T^{2} \)
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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

−10.91631442564475947143046809113, −10.02696520194473221335852241123, −9.452437323335425447901457942677, −8.297328202040022422151632478897, −7.53146206918284985831609850246, −6.48988613024097531621528338645, −5.20016873894121745159542275855, −5.00882266253020683080869222558, −4.17839469269585007904195327454, −1.77133720918587850686308692200, 0.04772586940799863856529105951, 1.09946125261145059378309427882, 2.62922458588015985534592220340, 4.34523876189797727893542388493, 5.26497643090550109170556745203, 6.33267032323396645952738075534, 7.20933600496806024081680630783, 7.84833064211373367281520700262, 8.362680233801646984783003256226, 10.30475917745644358519923124542

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