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Properties

Label 2-78-3.2-c4-0-12
Degree $2$
Conductor $78$
Sign $0.616 + 0.787i$
Analytic cond. $8.06285$
Root an. cond. $2.83951$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + (7.08 − 5.54i)3-s − 8.00·4-s − 20.4i·5-s + (15.6 + 20.0i)6-s − 5.83·7-s − 22.6i·8-s + (19.4 − 78.6i)9-s + 57.9·10-s − 148. i·11-s + (−56.7 + 44.3i)12-s − 46.8·13-s − 16.4i·14-s + (−113. − 145. i)15-s + 64.0·16-s − 155. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.787 − 0.616i)3-s − 0.500·4-s − 0.819i·5-s + (0.435 + 0.556i)6-s − 0.119·7-s − 0.353i·8-s + (0.240 − 0.970i)9-s + 0.579·10-s − 1.22i·11-s + (−0.393 + 0.308i)12-s − 0.277·13-s − 0.0841i·14-s + (−0.504 − 0.645i)15-s + 0.250·16-s − 0.536i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.616 + 0.787i$
Analytic conductor: \(8.06285\)
Root analytic conductor: \(2.83951\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :2),\ 0.616 + 0.787i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.67654 - 0.817003i\)
\(L(\frac12)\) \(\approx\) \(1.67654 - 0.817003i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + (-7.08 + 5.54i)T \)
13 \( 1 + 46.8T \)
good5 \( 1 + 20.4iT - 625T^{2} \)
7 \( 1 + 5.83T + 2.40e3T^{2} \)
11 \( 1 + 148. iT - 1.46e4T^{2} \)
17 \( 1 + 155. iT - 8.35e4T^{2} \)
19 \( 1 - 365.T + 1.30e5T^{2} \)
23 \( 1 - 231. iT - 2.79e5T^{2} \)
29 \( 1 - 516. iT - 7.07e5T^{2} \)
31 \( 1 - 896.T + 9.23e5T^{2} \)
37 \( 1 + 583.T + 1.87e6T^{2} \)
41 \( 1 - 559. iT - 2.82e6T^{2} \)
43 \( 1 + 2.86e3T + 3.41e6T^{2} \)
47 \( 1 - 3.62e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.47e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.69e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.99e3T + 1.38e7T^{2} \)
67 \( 1 - 5.57e3T + 2.01e7T^{2} \)
71 \( 1 + 2.73e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.40e3T + 2.83e7T^{2} \)
79 \( 1 + 1.76e3T + 3.89e7T^{2} \)
83 \( 1 + 4.62e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.48e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.67e4T + 8.85e7T^{2} \)
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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

−13.66002003076582776703021661179, −12.83120506917945451850139833902, −11.64935557449096038426352800111, −9.682109910310483881264667951973, −8.732027635618155797555246247094, −7.87366553106716663038999594974, −6.58185777260162303045549575834, −5.12586331265505575882512737537, −3.27579767434595350488485101277, −0.931616195656310809409016091407, 2.20810056790549219694975611209, 3.49245305864271894833203439745, 4.88464183839877183574269481107, 7.01826163531764689063108342529, 8.352876275582472724395661294788, 9.785075184265869165435374647481, 10.24233278816146251868008964916, 11.54437314943979349870874702813, 12.80909242491921997025894987982, 13.96047218279590163525470423036

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