Properties

Label 2-78-13.6-c2-0-1
Degree $2$
Conductor $78$
Sign $-0.679 - 0.733i$
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 + (−1.26 + 1.26i)5-s + (−2.36 − 0.633i)6-s + (−2.5 + 9.33i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (−2.19 − 1.26i)10-s + (9.92 − 2.66i)11-s − 3.46i·12-s + (−11.2 + 6.5i)13-s − 13.6·14-s + (−0.803 − 3i)15-s + (1.99 − 3.46i)16-s + (16.3 − 9.46i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.288 + 0.5i)3-s + (−0.433 + 0.250i)4-s + (−0.253 + 0.253i)5-s + (−0.394 − 0.105i)6-s + (−0.357 + 1.33i)7-s + (−0.250 − 0.249i)8-s + (−0.166 − 0.288i)9-s + (−0.219 − 0.126i)10-s + (0.902 − 0.241i)11-s − 0.288i·12-s + (−0.866 + 0.5i)13-s − 0.975·14-s + (−0.0535 − 0.200i)15-s + (0.124 − 0.216i)16-s + (0.964 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.679 - 0.733i$
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: \((2,\ 78,\ (\ :1),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.434034 + 0.993173i\)
\(L(\frac12)\) \(\approx\) \(0.434034 + 0.993173i\)
\(L(2)\) 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 \( 1 + (11.2 - 6.5i)T \)
good5 \( 1 + (1.26 - 1.26i)T - 25iT^{2} \)
7 \( 1 + (2.5 - 9.33i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-9.92 + 2.66i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-16.3 + 9.46i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-31.9 - 8.56i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (3.58 + 2.07i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.66 + 15i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-22.1 + 22.1i)T - 961iT^{2} \)
37 \( 1 + (35.7 - 9.58i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (6.67 + 24.9i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (13.2 - 7.66i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-7.01 - 7.01i)T + 2.20e3iT^{2} \)
53 \( 1 + 61.6T + 2.80e3T^{2} \)
59 \( 1 + (17.1 - 63.8i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-3.65 - 6.32i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.1 + 37.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-103. - 27.8i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (67.4 + 67.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 11.8T + 6.24e3T^{2} \)
83 \( 1 + (-111. + 111. i)T - 6.88e3iT^{2} \)
89 \( 1 + (-162. + 43.4i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-150. - 40.2i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−14.76258018103835137593754936262, −13.87063876470361311833954421286, −12.07810725960624377592000468622, −11.79187530280507001527892247511, −9.798512339003546038795027891924, −9.088788685155377069845353284056, −7.54636318421128527369071598232, −6.15013208642673797305223854709, −5.08331229238324740830885555784, −3.30530792085793444300787954630, 0.999668568583824310114476324065, 3.46449526572587331397000264109, 4.98287240467351646371301914262, 6.76798359684675156517611668642, 7.907370579766661779014210040975, 9.643585402789329734612986007535, 10.51641385646413912354650907864, 11.86964463421151892677111935353, 12.50376190549893572436274615880, 13.72029289088157699208296372213

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