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Properties

Label 2-76-19.4-c1-0-1
Degree $2$
Conductor $76$
Sign $0.314 + 0.949i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
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  + (−0.467 − 2.65i)3-s + (−0.982 + 0.824i)5-s + (1.67 − 2.89i)7-s + (−3.99 + 1.45i)9-s + (2.82 + 4.89i)11-s + (−0.0767 + 0.435i)13-s + (2.64 + 2.21i)15-s + (4.46 + 1.62i)17-s + (−4.08 + 1.53i)19-s + (−8.46 − 3.07i)21-s + (−2.81 − 2.36i)23-s + (−0.582 + 3.30i)25-s + (1.67 + 2.90i)27-s + (−2.83 + 1.03i)29-s + (5.39 − 9.33i)31-s + ⋯
L(s)  = 1  + (−0.269 − 1.53i)3-s + (−0.439 + 0.368i)5-s + (0.632 − 1.09i)7-s + (−1.33 + 0.484i)9-s + (0.852 + 1.47i)11-s + (−0.0212 + 0.120i)13-s + (0.682 + 0.572i)15-s + (1.08 + 0.394i)17-s + (−0.936 + 0.351i)19-s + (−1.84 − 0.671i)21-s + (−0.586 − 0.492i)23-s + (−0.116 + 0.660i)25-s + (0.322 + 0.559i)27-s + (−0.526 + 0.191i)29-s + (0.968 − 1.67i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704498 - 0.508668i\)
\(L(\frac12)\) \(\approx\) \(0.704498 - 0.508668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (4.08 - 1.53i)T \)
good3 \( 1 + (0.467 + 2.65i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (0.982 - 0.824i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.67 + 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.82 - 4.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0767 - 0.435i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.46 - 1.62i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.81 + 2.36i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.83 - 1.03i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.39 + 9.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + (-0.700 - 3.97i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.06 - 1.73i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (5.70 - 2.07i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.25 - 1.04i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (8.03 + 2.92i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.98 - 1.66i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.79 + 1.37i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (8.49 - 7.13i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.27 + 7.25i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.76 - 10.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.27 - 3.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.24 + 12.7i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-11.1 - 4.07i)T + (74.3 + 62.3i)T^{2} \)
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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

−14.27284048527809474231221344527, −13.08655590677361095819546160753, −12.18727515750706752877400521573, −11.32808414889007543111385480768, −9.982616806333446269240410368180, −7.972518953659119664447152712919, −7.34956190270354184596203296818, −6.33305183851324374639597987139, −4.22367855685489311720821965750, −1.64722581699495196770600832232, 3.49211598346924920243771243225, 4.88159172238965419349302016834, 5.94091772037393511552874018040, 8.359381968964710668770714904725, 9.021784935024921533547539632566, 10.33957918341980326663057157697, 11.46466358127401766876064517038, 12.09858638782125837835094542094, 14.02792437483337943285281703263, 14.93317316308305243452010504981

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