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Properties

Label 2-78-13.7-c2-0-3
Degree $2$
Conductor $78$
Sign $0.491 + 0.871i$
Analytic cond. $2.12534$
Root an. cond. $1.45785$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (0.866 − 1.5i)3-s + (1.73 − i)4-s + (−6.39 − 6.39i)5-s + (0.633 − 2.36i)6-s + (9.23 + 2.47i)7-s + (1.99 − 2i)8-s + (−1.5 − 2.59i)9-s + (−11.0 − 6.39i)10-s + (4.21 + 15.7i)11-s − 3.46i·12-s + (−3.81 + 12.4i)13-s + 13.5·14-s + (−15.1 + 4.05i)15-s + (1.99 − 3.46i)16-s + (8.31 − 4.80i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.288 − 0.5i)3-s + (0.433 − 0.250i)4-s + (−1.27 − 1.27i)5-s + (0.105 − 0.394i)6-s + (1.31 + 0.353i)7-s + (0.249 − 0.250i)8-s + (−0.166 − 0.288i)9-s + (−1.10 − 0.639i)10-s + (0.383 + 1.43i)11-s − 0.288i·12-s + (−0.293 + 0.956i)13-s + 0.965·14-s + (−1.00 + 0.270i)15-s + (0.124 − 0.216i)16-s + (0.489 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.491 + 0.871i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ 0.491 + 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55855 - 0.910359i\)
\(L(\frac12)\) \(\approx\) \(1.55855 - 0.910359i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
13 \( 1 + (3.81 - 12.4i)T \)
good5 \( 1 + (6.39 + 6.39i)T + 25iT^{2} \)
7 \( 1 + (-9.23 - 2.47i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-4.21 - 15.7i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-8.31 + 4.80i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.56 + 9.56i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (23.6 + 13.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13.8 - 23.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.5 - 11.5i)T + 961iT^{2} \)
37 \( 1 + (5.45 + 20.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-39.1 + 10.4i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (30.9 - 17.8i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-25.5 + 25.5i)T - 2.20e3iT^{2} \)
53 \( 1 + 39.0T + 2.80e3T^{2} \)
59 \( 1 + (46.0 + 12.3i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (35.2 + 61.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-38.7 + 10.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (8.28 - 30.9i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (9.68 - 9.68i)T - 5.32e3iT^{2} \)
79 \( 1 + 56.1T + 6.24e3T^{2} \)
83 \( 1 + (-4.59 - 4.59i)T + 6.88e3iT^{2} \)
89 \( 1 + (-29.9 - 111. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (9.04 - 33.7i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−14.12061860648960629168347880919, −12.53906028180152616612389977505, −12.14420016705047371443888095094, −11.33316130721313857469675100471, −9.281532057421380435377050040135, −8.121675137564520425107820526857, −7.17778594003674267697699602591, −4.99907588882584987091790229373, −4.23532752786441553675729096662, −1.71825785398380507253526867395, 3.18526005980408196591463513336, 4.17255629601728924615184353100, 5.89714304195550361863896705593, 7.70004399489160677635346583566, 8.098396779251393265756720560841, 10.40213509969278274872193023321, 11.22774547411602237019025045995, 11.89325531791146016990760858192, 13.79454768306024270841977814482, 14.49571305777956417182745109500

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