Properties

Label 2-78-39.32-c1-0-0
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

Related objects

Downloads

Learn more

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: $\chi_{78} (71, \cdot )$
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 \( 1 + (0.592 + 0.158i)T + (84.0 + 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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