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Properties

Label 2-76-76.75-c3-0-4
Degree $2$
Conductor $76$
Sign $-0.869 - 0.493i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 + 1.52i)2-s + 5.25·3-s + (3.36 − 7.25i)4-s − 19.3·5-s + (−12.5 + 7.98i)6-s + 22.0i·7-s + (3.00 + 22.4i)8-s + 0.564·9-s + (46.0 − 29.3i)10-s + 14.5i·11-s + (17.6 − 38.0i)12-s + 75.1i·13-s + (−33.6 − 52.6i)14-s − 101.·15-s + (−41.2 − 48.9i)16-s − 70.0·17-s + ⋯
L(s)  = 1  + (−0.842 + 0.537i)2-s + 1.01·3-s + (0.421 − 0.906i)4-s − 1.72·5-s + (−0.851 + 0.543i)6-s + 1.19i·7-s + (0.132 + 0.991i)8-s + 0.0209·9-s + (1.45 − 0.929i)10-s + 0.397i·11-s + (0.425 − 0.916i)12-s + 1.60i·13-s + (−0.641 − 1.00i)14-s − 1.74·15-s + (−0.645 − 0.764i)16-s − 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.161629 + 0.612102i\)
\(L(\frac12)\) \(\approx\) \(0.161629 + 0.612102i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.38 - 1.52i)T \)
19 \( 1 + (-6.74 + 82.5i)T \)
good3 \( 1 - 5.25T + 27T^{2} \)
5 \( 1 + 19.3T + 125T^{2} \)
7 \( 1 - 22.0iT - 343T^{2} \)
11 \( 1 - 14.5iT - 1.33e3T^{2} \)
13 \( 1 - 75.1iT - 2.19e3T^{2} \)
17 \( 1 + 70.0T + 4.91e3T^{2} \)
23 \( 1 - 50.4iT - 1.21e4T^{2} \)
29 \( 1 + 186. iT - 2.43e4T^{2} \)
31 \( 1 - 194.T + 2.97e4T^{2} \)
37 \( 1 - 225. iT - 5.06e4T^{2} \)
41 \( 1 - 35.2iT - 6.89e4T^{2} \)
43 \( 1 + 11.1iT - 7.95e4T^{2} \)
47 \( 1 - 365. iT - 1.03e5T^{2} \)
53 \( 1 - 53.8iT - 1.48e5T^{2} \)
59 \( 1 - 525.T + 2.05e5T^{2} \)
61 \( 1 + 235.T + 2.26e5T^{2} \)
67 \( 1 + 80.5T + 3.00e5T^{2} \)
71 \( 1 + 362.T + 3.57e5T^{2} \)
73 \( 1 + 331.T + 3.89e5T^{2} \)
79 \( 1 - 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + 247. iT - 5.71e5T^{2} \)
89 \( 1 - 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 - 438. iT - 9.12e5T^{2} \)
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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

−15.05717797467430017228064529580, −13.73837859013021522894890721727, −11.83592997102689700362023742437, −11.37452497770509054032981218883, −9.349985123695381491768708704008, −8.703520588374495142382479304394, −7.84121993548433216974101069931, −6.64496387754500882446616219069, −4.50066085471717418686924430578, −2.50017354380640151043460208046, 0.45829124869701688400752786296, 3.13329346512263953813243692477, 3.98552167685599122143881501663, 7.20238912124034445483467250990, 8.039771102912898329157739596880, 8.623372472512988046294676333250, 10.33849274421578335807419856111, 11.10571003833926914936499175887, 12.32849216635438527508697058781, 13.41255674532431258134305558623

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