Properties

Label 2-78-39.5-c1-0-1
Degree $2$
Conductor $78$
Sign $0.377 - 0.925i$
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  + (0.707 + 0.707i)2-s + (−1.64 + 0.542i)3-s + 1.00i·4-s + (2.32 + 2.32i)5-s + (−1.54 − 0.779i)6-s + (−1.76 − 1.76i)7-s + (−0.707 + 0.707i)8-s + (2.41 − 1.78i)9-s + 3.28i·10-s + (1.08 − 1.08i)11-s + (−0.542 − 1.64i)12-s + (0.766 − 3.52i)13-s − 2.49i·14-s + (−5.08 − 2.56i)15-s − 1.00·16-s − 5.73·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.949 + 0.313i)3-s + 0.500i·4-s + (1.04 + 1.04i)5-s + (−0.631 − 0.318i)6-s + (−0.667 − 0.667i)7-s + (−0.250 + 0.250i)8-s + (0.803 − 0.594i)9-s + 1.04i·10-s + (0.327 − 0.327i)11-s + (−0.156 − 0.474i)12-s + (0.212 − 0.977i)13-s − 0.667i·14-s + (−1.31 − 0.662i)15-s − 0.250·16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.377 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.836802 + 0.562410i\)
\(L(\frac12)\) \(\approx\) \(0.836802 + 0.562410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.64 - 0.542i)T \)
13 \( 1 + (-0.766 + 3.52i)T \)
good5 \( 1 + (-2.32 - 2.32i)T + 5iT^{2} \)
7 \( 1 + (1.76 + 1.76i)T + 7iT^{2} \)
11 \( 1 + (-1.08 + 1.08i)T - 11iT^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (-2.28 + 2.28i)T - 19iT^{2} \)
23 \( 1 - 4.65T + 23T^{2} \)
29 \( 1 - 4.65iT - 29T^{2} \)
31 \( 1 + (3.82 - 3.82i)T - 31iT^{2} \)
37 \( 1 + (3.05 + 3.05i)T + 37iT^{2} \)
41 \( 1 + (0.410 + 0.410i)T + 41iT^{2} \)
43 \( 1 - 0.222iT - 43T^{2} \)
47 \( 1 + (7.65 - 7.65i)T - 47iT^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 + (4.65 - 4.65i)T - 59iT^{2} \)
61 \( 1 - 3.06T + 61T^{2} \)
67 \( 1 + (-0.533 + 0.533i)T - 67iT^{2} \)
71 \( 1 + (5.48 + 5.48i)T + 71iT^{2} \)
73 \( 1 + (-2.28 - 2.28i)T + 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (-1.39 - 1.39i)T + 83iT^{2} \)
89 \( 1 + (6.41 - 6.41i)T - 89iT^{2} \)
97 \( 1 + (11.5 - 11.5i)T - 97iT^{2} \)
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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

−14.68978640748632422889247770804, −13.52954049974063275935111558051, −12.83025342833026939793534517856, −11.14182548223792799006200368299, −10.49548947113355975484018925098, −9.264623025758729573648282848733, −7.00162012334420338726415072711, −6.43355932617564198985975048687, −5.19456813851508398182693589493, −3.35316169535355541291048359399, 1.85229661636046153072729210994, 4.56914739323237194620808803590, 5.71961690734794513396796625714, 6.66038575585206832989404981494, 9.023275389283640350843330541096, 9.797742668362017065530409415905, 11.26789805813860441883792601565, 12.22988529884159221146279870959, 13.05239737227398673457691208478, 13.70985641281234945737565128804

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