Properties

Label 2-78-39.8-c1-0-0
Degree $2$
Conductor $78$
Sign $-0.854 - 0.519i$
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 + (0.779 − 1.54i)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 + (2.56 − 5.08i)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 + (0.318 − 0.631i)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 + (0.662 − 1.31i)15-s − 0.250·16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.854 - 0.519i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ -0.854 - 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101482 + 0.362012i\)
\(L(\frac12)\) \(\approx\) \(0.101482 + 0.362012i\)
\(L(\frac{3}{2})\) 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

−15.21559629337564665251279424210, −14.21437097418132054823622708361, −12.33369294520362025206555627566, −11.53495239857141919473010075133, −10.50437701126341711412792696544, −9.458727995877703871432481256466, −7.80855939658863933736575452399, −6.70106682838207334307959722476, −5.62711129380329807393199177493, −3.64454303557230889729777468219, 0.65791028884537277568242105239, 3.85016665786878510166097608901, 5.38877139318244346406308799767, 7.28821259400927405137656192568, 8.116250488168503621611169517627, 9.788986274779415543216189589009, 10.73038231481550192999613296103, 12.02781822539046693838792046587, 12.49770563619411992084178629694, 13.49661036847863529853983047165

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