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Properties

Label 2-78-13.3-c1-0-1
Degree $2$
Conductor $78$
Sign $0.872 - 0.488i$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s + (0.499 − 0.866i)6-s + (−1 + 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)10-s + (−3 − 5.19i)11-s + 0.999·12-s + (−3.5 + 0.866i)13-s − 1.99·14-s + (−1.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (0.204 − 0.353i)6-s + (−0.377 + 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (−0.904 − 1.56i)11-s + 0.288·12-s + (−0.970 + 0.240i)13-s − 0.534·14-s + (−0.387 − 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05333 + 0.275014i\)
\(L(\frac12)\) \(\approx\) \(1.05333 + 0.275014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (3.5 - 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.27867396311595024572434941436, −13.51817964603147958594568172442, −12.76007350342529018271265712239, −11.46457703036581464797198554723, −9.948149548125888805029333159835, −8.819409285103628851972636338272, −7.36944870964190706456148621822, −5.89970760623753785312771422036, −5.45151851794710074846736962050, −2.71535848121464506175339869070, 2.41599769197848907918756184175, 4.50903381537671760984673464703, 5.63234435803273217669811201665, 7.16326943265698509724527658031, 9.336707253707236921592227844229, 10.13411025138224755705732726398, 10.69811513121485893252081626758, 12.56867437975354614470896300496, 12.98706950235938947359996027931, 14.35286686074572368409181867512

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