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Properties

Label 2-78-3.2-c2-0-2
Degree $2$
Conductor $78$
Sign $0.919 - 0.393i$
Analytic cond. $2.12534$
Root an. cond. $1.45785$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (1.18 + 2.75i)3-s − 2.00·4-s + 5.25i·5-s + (3.90 − 1.66i)6-s + 8.86·7-s + 2.82i·8-s + (−6.21 + 6.51i)9-s + 7.43·10-s − 6.35i·11-s + (−2.36 − 5.51i)12-s + 3.60·13-s − 12.5i·14-s + (−14.5 + 6.21i)15-s + 4.00·16-s − 22.6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.393 + 0.919i)3-s − 0.500·4-s + 1.05i·5-s + (0.650 − 0.278i)6-s + 1.26·7-s + 0.353i·8-s + (−0.690 + 0.723i)9-s + 0.743·10-s − 0.577i·11-s + (−0.196 − 0.459i)12-s + 0.277·13-s − 0.895i·14-s + (−0.967 + 0.414i)15-s + 0.250·16-s − 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ 0.919 - 0.393i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39811 + 0.286739i\)
\(L(\frac12)\) \(\approx\) \(1.39811 + 0.286739i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-1.18 - 2.75i)T \)
13 \( 1 - 3.60T \)
good5 \( 1 - 5.25iT - 25T^{2} \)
7 \( 1 - 8.86T + 49T^{2} \)
11 \( 1 + 6.35iT - 121T^{2} \)
17 \( 1 + 22.6iT - 289T^{2} \)
19 \( 1 + 9.32T + 361T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 + 43.0iT - 841T^{2} \)
31 \( 1 + 53.0T + 961T^{2} \)
37 \( 1 - 30.3T + 1.36e3T^{2} \)
41 \( 1 - 9.56iT - 1.68e3T^{2} \)
43 \( 1 - 67.2T + 1.84e3T^{2} \)
47 \( 1 + 50.3iT - 2.20e3T^{2} \)
53 \( 1 + 41.6iT - 2.80e3T^{2} \)
59 \( 1 - 43.3iT - 3.48e3T^{2} \)
61 \( 1 - 106.T + 3.72e3T^{2} \)
67 \( 1 + 45.4T + 4.48e3T^{2} \)
71 \( 1 - 48.0iT - 5.04e3T^{2} \)
73 \( 1 + 123.T + 5.32e3T^{2} \)
79 \( 1 + 94.8T + 6.24e3T^{2} \)
83 \( 1 - 108. iT - 6.88e3T^{2} \)
89 \( 1 - 42.2iT - 7.92e3T^{2} \)
97 \( 1 + 88.5T + 9.40e3T^{2} \)
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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

−14.35027496114927595257622218631, −13.48020970822303525810406528242, −11.38206629258418182833801508337, −11.20255381225410794923934038796, −10.01089853780307069908032608531, −8.825048309695312513918574618511, −7.59936561196509219705317999188, −5.45619291949927174002570989335, −4.00241234720404401747900686026, −2.55398085884271648321791957207, 1.54247033246985484347967499447, 4.40883361872324736331240905829, 5.79247265151645748307336449875, 7.32366640960686241170153974185, 8.358849709146284464365362287262, 8.969399383791618562012548824430, 10.93406335793213740422425826372, 12.51823417598093269329672273575, 12.89869316232090165161149209510, 14.40173499032361609016850028388

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