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Properties

Label 2-78-39.11-c1-0-1
Degree $2$
Conductor $78$
Sign $0.983 + 0.179i$
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 + (−0.313 + 0.313i)5-s + (−1.69 − 0.370i)6-s + (−0.0745 − 0.278i)7-s + (−0.707 − 0.707i)8-s + (2.98 − 0.275i)9-s + (0.383 − 0.221i)10-s + (−0.150 + 0.563i)11-s + (1.53 + 0.796i)12-s + (−1.79 − 3.12i)13-s + 0.288i·14-s + (−0.517 + 0.567i)15-s + (0.500 + 0.866i)16-s + (−2.79 + 4.84i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.998 − 0.0459i)3-s + (0.433 + 0.249i)4-s + (−0.140 + 0.140i)5-s + (−0.690 − 0.151i)6-s + (−0.0281 − 0.105i)7-s + (−0.249 − 0.249i)8-s + (0.995 − 0.0917i)9-s + (0.121 − 0.0700i)10-s + (−0.0454 + 0.169i)11-s + (0.444 + 0.229i)12-s + (−0.496 − 0.867i)13-s + 0.0770i·14-s + (−0.133 + 0.146i)15-s + (0.125 + 0.216i)16-s + (−0.678 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.983 + 0.179i$
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: \((2,\ 78,\ (\ :1/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897149 - 0.0813565i\)
\(L(\frac12)\) \(\approx\) \(0.897149 - 0.0813565i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 3.12i)T \)
good5 \( 1 + (0.313 - 0.313i)T - 5iT^{2} \)
7 \( 1 + (0.0745 + 0.278i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.150 - 0.563i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.79 - 4.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.79 - 1.81i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.32 + 5.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.57 + 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 + 1.03i)T + 31iT^{2} \)
37 \( 1 + (-6.72 - 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.36 - 1.97i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.26 + 1.88i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.71 + 3.71i)T + 47iT^{2} \)
53 \( 1 + 3.64iT - 53T^{2} \)
59 \( 1 + (-3.29 + 0.881i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.25 - 9.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.29 + 8.55i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.98 - 14.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.52 + 3.52i)T - 73iT^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + (-8.23 + 8.23i)T - 83iT^{2} \)
89 \( 1 + (3.64 - 13.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.59 + 0.694i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.77228165557714755869520130542, −13.18706776357299593689498090174, −12.43989411568034970309111482560, −10.75266794244188091642774955194, −9.965867861437631100765269549410, −8.613053818301682148318463255976, −7.88656003652607058517979901626, −6.49996029464385801436391250431, −4.07641749843765235876016347203, −2.34275675523666473044846469699, 2.39449528467691720530944554359, 4.44211712062665931512074477069, 6.60858545746184867538943732661, 7.77532382286985354468170950158, 8.895108233900281771634528627451, 9.646254299324990271718437160140, 11.00719039803608678476597237266, 12.32927337919761791092082365974, 13.64708313608005216943725208744, 14.55299505727916212341205562331

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