Properties

Label 2-76-76.7-c2-0-7
Degree $2$
Conductor $76$
Sign $-0.0265 + 0.999i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
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  + (−1.93 − 0.489i)2-s + (−3.88 + 2.24i)3-s + (3.52 + 1.89i)4-s + (−0.133 − 0.231i)5-s + (8.63 − 2.44i)6-s − 7.24i·7-s + (−5.89 − 5.40i)8-s + (5.56 − 9.64i)9-s + (0.145 + 0.514i)10-s − 11.7i·11-s + (−17.9 + 0.519i)12-s + (4.17 − 7.22i)13-s + (−3.54 + 14.0i)14-s + (1.03 + 0.600i)15-s + (8.78 + 13.3i)16-s + (−7.11 − 12.3i)17-s + ⋯
L(s)  = 1  + (−0.969 − 0.244i)2-s + (−1.29 + 0.747i)3-s + (0.880 + 0.474i)4-s + (−0.0267 − 0.0463i)5-s + (1.43 − 0.408i)6-s − 1.03i·7-s + (−0.737 − 0.675i)8-s + (0.618 − 1.07i)9-s + (0.0145 + 0.0514i)10-s − 1.06i·11-s + (−1.49 + 0.0433i)12-s + (0.320 − 0.555i)13-s + (−0.253 + 1.00i)14-s + (0.0693 + 0.0400i)15-s + (0.549 + 0.835i)16-s + (−0.418 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0265 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0265 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0265 + 0.999i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.0265 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.268820 - 0.276056i\)
\(L(\frac12)\) \(\approx\) \(0.268820 - 0.276056i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.93 + 0.489i)T \)
19 \( 1 + (-11.9 - 14.7i)T \)
good3 \( 1 + (3.88 - 2.24i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.133 + 0.231i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 7.24iT - 49T^{2} \)
11 \( 1 + 11.7iT - 121T^{2} \)
13 \( 1 + (-4.17 + 7.22i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (7.11 + 12.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (21.4 + 12.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (25.7 - 44.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 26.3iT - 961T^{2} \)
37 \( 1 + 20.1T + 1.36e3T^{2} \)
41 \( 1 + (38.5 + 66.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.0 - 17.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-31.9 - 18.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (8.75 - 15.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-82.8 + 47.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-26.2 + 45.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.8 - 6.86i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-39.7 + 22.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-20.0 - 34.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (106. - 61.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 108. iT - 6.88e3T^{2} \)
89 \( 1 + (26.2 - 45.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-29.9 - 51.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.98635235188784667805321943426, −12.43098505579967710892673205753, −11.30554406164869125948752433230, −10.67580662105441235515980733401, −9.882746222346983321379308114575, −8.370561183991821661824789214190, −6.89779773245301103268330142000, −5.59256319662274036569281796490, −3.74336430207514866287306716890, −0.51992516672942714813786606603, 1.82909899905792982644583015734, 5.34081728992338719096848770396, 6.40333141509512835430284510174, 7.36529896733971485470073632394, 8.850481450406695520391763365467, 10.08952004363684278732651895820, 11.50659592437405554817870573439, 11.86405791842542884601835379181, 13.12571194172529551554423245953, 14.97489568206873919578035143634

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