Properties

Label 2-78-39.11-c1-0-2
Degree $2$
Conductor $78$
Sign $0.503 + 0.863i$
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 + (−1.73 + 0.0795i)3-s + (0.866 + 0.499i)4-s + (2.76 − 2.76i)5-s + (1.69 + 0.370i)6-s + (−0.657 − 2.45i)7-s + (−0.707 − 0.707i)8-s + (2.98 − 0.275i)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 + (−4.56 + 5.00i)15-s + (0.500 + 0.866i)16-s + (−0.547 + 0.947i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.998 + 0.0459i)3-s + (0.433 + 0.249i)4-s + (1.23 − 1.23i)5-s + (0.690 + 0.151i)6-s + (−0.248 − 0.927i)7-s + (−0.249 − 0.249i)8-s + (0.995 − 0.0917i)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 + (−1.17 + 1.29i)15-s + (0.125 + 0.216i)16-s + (−0.132 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.503 + 0.863i$
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: \((2,\ 78,\ (\ :1/2),\ 0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541195 - 0.310822i\)
\(L(\frac12)\) \(\approx\) \(0.541195 - 0.310822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0795i)T \)
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

−13.96990827521239163129969228716, −13.01094577417330579719353119279, −12.08291390960390321617270206216, −10.77437535999277652850603037968, −9.822415684090975277798845560731, −8.987969982666816580414105096099, −7.17664124224794375165555332761, −5.93778353271788793564928047819, −4.55530097555059560367388182088, −1.34656145514284277426540411701, 2.45921500305664462489874584101, 5.56118808710998277745743518928, 6.26254054257138010928345051105, 7.44950739516112873005107145067, 9.404058646839904160017379657961, 10.18044721148571706897582320190, 11.07327464186337698655231063641, 12.27624426484361489416134345843, 13.54546286287021020869111194174, 14.94854704435803067188111040767

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