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Properties

Label 2-78-39.11-c1-0-3
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: $\chi_{78} (11, \cdot )$
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

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

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