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Properties

Label 2-78-39.32-c1-0-2
Degree $2$
Conductor $78$
Sign $0.996 - 0.0872i$
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  + (0.965 − 0.258i)2-s + (0.933 + 1.45i)3-s + (0.866 − 0.499i)4-s + (−2.76 − 2.76i)5-s + (1.27 + 1.16i)6-s + (−0.657 + 2.45i)7-s + (0.707 − 0.707i)8-s + (−1.25 + 2.72i)9-s + (−3.38 − 1.95i)10-s + (−0.150 − 0.563i)11-s + (1.53 + 0.796i)12-s + (−1.20 − 3.39i)13-s + 2.54i·14-s + (1.44 − 6.61i)15-s + (0.500 − 0.866i)16-s + (0.547 + 0.947i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.539 + 0.842i)3-s + (0.433 − 0.249i)4-s + (−1.23 − 1.23i)5-s + (0.522 + 0.476i)6-s + (−0.248 + 0.927i)7-s + (0.249 − 0.249i)8-s + (−0.418 + 0.908i)9-s + (−1.07 − 0.617i)10-s + (−0.0454 − 0.169i)11-s + (0.444 + 0.229i)12-s + (−0.335 − 0.942i)13-s + 0.678i·14-s + (0.374 − 1.70i)15-s + (0.125 − 0.216i)16-s + (0.132 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.996 - 0.0872i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.996 - 0.0872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28458 + 0.0561330i\)
\(L(\frac12)\) \(\approx\) \(1.28458 + 0.0561330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.933 - 1.45i)T \)
13 \( 1 + (1.20 + 3.39i)T \)
good5 \( 1 + (2.76 + 2.76i)T + 5iT^{2} \)
7 \( 1 + (0.657 - 2.45i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.150 + 0.563i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.547 - 0.947i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 0.355i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.876 - 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.12 - 2.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.49 + 6.49i)T - 31iT^{2} \)
37 \( 1 + (2.98 - 0.801i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.11 - 1.37i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.26 + 1.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.51 - 5.51i)T - 47iT^{2} \)
53 \( 1 + 3.04iT - 53T^{2} \)
59 \( 1 + (-8.19 - 2.19i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.67 + 8.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.70 - 6.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.220 + 0.821i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.18 + 5.18i)T + 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (5.15 + 5.15i)T + 83iT^{2} \)
89 \( 1 + (2.50 + 9.35i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.592 + 0.158i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.69538724394213686996021682596, −13.26908012586179687170283393777, −12.31755448242819468788746274486, −11.49830683654614798914800298247, −10.01108643695508813791653489372, −8.725061249331501625610654408497, −7.892186164221655801453665623808, −5.52110308212129519907950328248, −4.50056986383673889244168068547, −3.14620252711844967410973681363, 2.94306517008290312810155107223, 4.14983643549329667687616657811, 6.72470149759890707001048170868, 7.11357907081987414385645787302, 8.243193406660291654302317906166, 10.23774730761562135105749531011, 11.55730181339464558507290548531, 12.23095785628074332237092662983, 13.71360530526999302160458499805, 14.24900524639273940795377183105

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