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Properties

Label 2-78-13.4-c1-0-3
Degree $2$
Conductor $78$
Sign $0.997 - 0.0771i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
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 + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − 1.73i·5-s + (0.866 − 0.499i)6-s + (−4.09 + 2.36i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.49i)10-s + (4.09 + 2.36i)11-s + 0.999·12-s + (−3.59 + 0.232i)13-s − 4.73·14-s + (−1.49 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−2.59 − 4.5i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s − 0.774i·5-s + (0.353 − 0.204i)6-s + (−1.54 + 0.894i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.273 − 0.474i)10-s + (1.23 + 0.713i)11-s + 0.288·12-s + (−0.997 + 0.0643i)13-s − 1.26·14-s + (−0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.997 - 0.0771i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.997 - 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24124 + 0.0479530i\)
\(L(\frac12)\) \(\approx\) \(1.24124 + 0.0479530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (3.59 - 0.232i)T \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + (4.09 - 2.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.401 - 0.232i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.09 - 5.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.26iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (12 - 6.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.40 + 4.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.29 + 5.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.43106666018703551964040024444, −13.24434780089022654718595896838, −12.42864789776079111620075164178, −11.89914697614154461698159110569, −9.552376313823090477480147145279, −8.950812568288712034051387321784, −7.17877352072917162392944041289, −6.26484740831525931551073563134, −4.67328986939593676657511962972, −2.78365226814312808697877704458, 3.10639155598610243462545224339, 4.05794074943596429287036818958, 6.15874339313983288500300342785, 7.10699435434673673449135277964, 9.184375554998852900409044871602, 10.17494998809962064716079754732, 11.00628850508995611589443769635, 12.40716194884352239996051409917, 13.50523769679742012855965949556, 14.35055208016376764816169793016

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