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Properties

Label 2-78-39.8-c1-0-3
Degree $2$
Conductor $78$
Sign $0.904 + 0.427i$
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 + (0.352 − 1.69i)3-s − 1.00i·4-s + (0.499 − 0.499i)5-s + (0.949 + 1.44i)6-s + (1.39 − 1.39i)7-s + (0.707 + 0.707i)8-s + (−2.75 − 1.19i)9-s + 0.705i·10-s + (3.39 + 3.39i)11-s + (−1.69 − 0.352i)12-s + (−2.39 + 2.69i)13-s + 1.97i·14-s + (−0.670 − 1.02i)15-s − 1.00·16-s − 4.38·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.203 − 0.979i)3-s − 0.500i·4-s + (0.223 − 0.223i)5-s + (0.387 + 0.591i)6-s + (0.528 − 0.528i)7-s + (0.250 + 0.250i)8-s + (−0.916 − 0.398i)9-s + 0.223i·10-s + (1.02 + 1.02i)11-s + (−0.489 − 0.101i)12-s + (−0.665 + 0.746i)13-s + 0.528i·14-s + (−0.173 − 0.263i)15-s − 0.250·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.904 + 0.427i$
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: \((2,\ 78,\ (\ :1/2),\ 0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809677 - 0.181661i\)
\(L(\frac12)\) \(\approx\) \(0.809677 - 0.181661i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.352 + 1.69i)T \)
13 \( 1 + (2.39 - 2.69i)T \)
good5 \( 1 + (-0.499 + 0.499i)T - 5iT^{2} \)
7 \( 1 + (-1.39 + 1.39i)T - 7iT^{2} \)
11 \( 1 + (-3.39 - 3.39i)T + 11iT^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 + (1.70 + 1.70i)T + 19iT^{2} \)
23 \( 1 - 0.998T + 23T^{2} \)
29 \( 1 + 0.998iT - 29T^{2} \)
31 \( 1 + (-6.50 - 6.50i)T + 31iT^{2} \)
37 \( 1 + (-4.10 + 4.10i)T - 37iT^{2} \)
41 \( 1 + (5.24 - 5.24i)T - 41iT^{2} \)
43 \( 1 + 8.88iT - 43T^{2} \)
47 \( 1 + (-0.352 - 0.352i)T + 47iT^{2} \)
53 \( 1 + 14.2iT - 53T^{2} \)
59 \( 1 + (0.998 + 0.998i)T + 59iT^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 + (5.79 + 5.79i)T + 67iT^{2} \)
71 \( 1 + (-7.13 + 7.13i)T - 71iT^{2} \)
73 \( 1 + (1.70 - 1.70i)T - 73iT^{2} \)
79 \( 1 + 0.207T + 79T^{2} \)
83 \( 1 + (9.17 - 9.17i)T - 83iT^{2} \)
89 \( 1 + (2.54 + 2.54i)T + 89iT^{2} \)
97 \( 1 + (3.58 + 3.58i)T + 97iT^{2} \)
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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

−14.37845719444423350492558052779, −13.49933838011291655543387444188, −12.24929209545029960934420765931, −11.17827947322191002516279624220, −9.563106635356523193688611300644, −8.618245454950493986541791259070, −7.25067094535130140011454523132, −6.59233328333973322914896957959, −4.67785942395056560720689591604, −1.78387708746019777554034924115, 2.73922884705104817990920566359, 4.40623838732767721349959991332, 6.09329546592661980772416194972, 8.165226250506851406963132276306, 8.997530223890113821533211005259, 10.10520054462771031350324339871, 11.10710181673387832485079686117, 11.96042209934546199123513278157, 13.57549953534837457635109070164, 14.65127746436878378101886415441

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