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Properties

Label 2-78-13.4-c1-0-0
Degree $2$
Conductor $78$
Sign $0.711 - 0.702i$
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 + 3.73i·5-s + (0.866 − 0.499i)6-s + (2.36 − 1.36i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.86 − 3.23i)10-s + (1.09 + 0.633i)11-s − 0.999·12-s + (−2.59 + 2.5i)13-s − 2.73·14-s + (−3.23 − 1.86i)15-s + (−0.5 + 0.866i)16-s + (−2.86 − 4.96i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 1.66i·5-s + (0.353 − 0.204i)6-s + (0.894 − 0.516i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.590 − 1.02i)10-s + (0.331 + 0.191i)11-s − 0.288·12-s + (−0.720 + 0.693i)13-s − 0.730·14-s + (−0.834 − 0.481i)15-s + (−0.125 + 0.216i)16-s + (−0.695 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.711 - 0.702i$
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: \((2,\ 78,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650993 + 0.267208i\)
\(L(\frac12)\) \(\approx\) \(0.650993 + 0.267208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good5 \( 1 - 3.73iT - 5T^{2} \)
7 \( 1 + (-2.36 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-3.06 - 1.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.13 - 4.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.83 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.3 + 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 2.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 0.196iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.73i)T + (44.5 + 77.0i)T^{2} \)
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

−14.58726237725523683378996541707, −13.83180864656526398757198909823, −11.74771978406908958698379363775, −11.23304049319163590498207092416, −10.30144854453387930859462299316, −9.292661793350265110313610425119, −7.51424907229592473310621805340, −6.70868088619909623586300094746, −4.53422574343959597122805498660, −2.72857001623338863597294091313, 1.45567106767391687444528325286, 4.87262416371336868153476395314, 5.84584059943671257987957986023, 7.72992778196790332747013034511, 8.485413756317002429117953726751, 9.513381129697581069675827652353, 11.15005717280331496866148679860, 12.21687005393779013015749045059, 13.01303943299352064092523889288, 14.43499675835519792234833728187

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