Properties

Label 2-78-13.6-c2-0-2
Degree $2$
Conductor $78$
Sign $0.327 + 0.944i$
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  + (−0.366 − 1.36i)2-s + (−0.866 + 1.5i)3-s + (−1.73 + i)4-s + (4.41 − 4.41i)5-s + (2.36 + 0.633i)6-s + (2.11 − 7.89i)7-s + (2 + 1.99i)8-s + (−1.5 − 2.59i)9-s + (−7.64 − 4.41i)10-s + (18.5 − 4.96i)11-s − 3.46i·12-s + (−12.9 + 1.42i)13-s − 11.5·14-s + (2.79 + 10.4i)15-s + (1.99 − 3.46i)16-s + (−19.9 + 11.5i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.288 + 0.5i)3-s + (−0.433 + 0.250i)4-s + (0.882 − 0.882i)5-s + (0.394 + 0.105i)6-s + (0.302 − 1.12i)7-s + (0.250 + 0.249i)8-s + (−0.166 − 0.288i)9-s + (−0.764 − 0.441i)10-s + (1.68 − 0.451i)11-s − 0.288i·12-s + (−0.993 + 0.109i)13-s − 0.825·14-s + (0.186 + 0.696i)15-s + (0.124 − 0.216i)16-s + (−1.17 + 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ 0.327 + 0.944i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.959235 - 0.683017i\)
\(L(\frac12)\) \(\approx\) \(0.959235 - 0.683017i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
13 \( 1 + (12.9 - 1.42i)T \)
good5 \( 1 + (-4.41 + 4.41i)T - 25iT^{2} \)
7 \( 1 + (-2.11 + 7.89i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-18.5 + 4.96i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (19.9 - 11.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.417 - 0.111i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-36.4 - 21.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (3.25 - 5.64i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (17.8 - 17.8i)T - 961iT^{2} \)
37 \( 1 + (1.37 - 0.369i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (10.8 + 40.3i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (51.1 - 29.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-15.0 - 15.0i)T + 2.20e3iT^{2} \)
53 \( 1 - 8.90T + 2.80e3T^{2} \)
59 \( 1 + (-11.4 + 42.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-44.8 - 77.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.2 - 38.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-8.56 - 2.29i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (5.92 + 5.92i)T + 5.32e3iT^{2} \)
79 \( 1 - 115.T + 6.24e3T^{2} \)
83 \( 1 + (34.2 - 34.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (-58.0 + 15.5i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (129. + 34.6i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−13.80009031915004076243281517700, −12.90138915821490620126516335883, −11.62723073396572357982864676245, −10.68289343287142191293046141659, −9.493986630827335430761874838348, −8.842022789598711718916878496088, −6.87258450767649471244018423052, −5.11250641520283181732055064418, −3.93440274553145140442271240797, −1.34172481331503456514989208179, 2.26833808309964379848734156608, 5.01986733385761207909467476060, 6.41204933831995675619945779618, 7.01263768164125617316595385394, 8.814811929872529475379449746033, 9.672501910689072450632794678935, 11.19871852399837706516262635719, 12.26183401764840487921980219900, 13.55290292632246701305069026440, 14.70421119198443169599217570972

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