Properties

Label 2-76-19.13-c4-0-5
Degree $2$
Conductor $76$
Sign $-0.135 + 0.990i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.65 − 7.29i)3-s + (0.405 + 2.29i)5-s + (23.2 − 40.3i)7-s + (15.9 + 13.3i)9-s + (−88.3 − 152. i)11-s + (−37.7 − 103. i)13-s + (17.8 + 3.14i)15-s + (−57.1 + 47.9i)17-s + (−192. − 305. i)19-s + (−232. − 276. i)21-s + (15.6 − 88.7i)23-s + (582. − 211. i)25-s + (684. − 394. i)27-s + (−121. + 145. i)29-s + (670. + 387. i)31-s + ⋯
L(s)  = 1  + (0.294 − 0.810i)3-s + (0.0162 + 0.0919i)5-s + (0.474 − 0.822i)7-s + (0.196 + 0.165i)9-s + (−0.729 − 1.26i)11-s + (−0.223 − 0.613i)13-s + (0.0792 + 0.0139i)15-s + (−0.197 + 0.165i)17-s + (−0.533 − 0.845i)19-s + (−0.526 − 0.627i)21-s + (0.0295 − 0.167i)23-s + (0.931 − 0.339i)25-s + (0.938 − 0.541i)27-s + (−0.144 + 0.172i)29-s + (0.697 + 0.402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.13042 - 1.29615i\)
\(L(\frac12)\) \(\approx\) \(1.13042 - 1.29615i\)
\(L(3)\) 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 \( 1 + (192. + 305. i)T \)
good3 \( 1 + (-2.65 + 7.29i)T + (-62.0 - 52.0i)T^{2} \)
5 \( 1 + (-0.405 - 2.29i)T + (-587. + 213. i)T^{2} \)
7 \( 1 + (-23.2 + 40.3i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (88.3 + 152. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (37.7 + 103. i)T + (-2.18e4 + 1.83e4i)T^{2} \)
17 \( 1 + (57.1 - 47.9i)T + (1.45e4 - 8.22e4i)T^{2} \)
23 \( 1 + (-15.6 + 88.7i)T + (-2.62e5 - 9.57e4i)T^{2} \)
29 \( 1 + (121. - 145. i)T + (-1.22e5 - 6.96e5i)T^{2} \)
31 \( 1 + (-670. - 387. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.35e3iT - 1.87e6T^{2} \)
41 \( 1 + (316. - 870. i)T + (-2.16e6 - 1.81e6i)T^{2} \)
43 \( 1 + (-298. - 1.69e3i)T + (-3.21e6 + 1.16e6i)T^{2} \)
47 \( 1 + (-1.73e3 - 1.45e3i)T + (8.47e5 + 4.80e6i)T^{2} \)
53 \( 1 + (1.14e3 + 201. i)T + (7.41e6 + 2.69e6i)T^{2} \)
59 \( 1 + (-2.59e3 - 3.08e3i)T + (-2.10e6 + 1.19e7i)T^{2} \)
61 \( 1 + (-702. + 3.98e3i)T + (-1.30e7 - 4.73e6i)T^{2} \)
67 \( 1 + (-1.78e3 + 2.12e3i)T + (-3.49e6 - 1.98e7i)T^{2} \)
71 \( 1 + (-5.74e3 + 1.01e3i)T + (2.38e7 - 8.69e6i)T^{2} \)
73 \( 1 + (4.48e3 + 1.63e3i)T + (2.17e7 + 1.82e7i)T^{2} \)
79 \( 1 + (-1.60e3 + 4.40e3i)T + (-2.98e7 - 2.50e7i)T^{2} \)
83 \( 1 + (3.71e3 - 6.42e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (1.54e3 + 4.25e3i)T + (-4.80e7 + 4.03e7i)T^{2} \)
97 \( 1 + (-3.36e3 - 4.01e3i)T + (-1.53e7 + 8.71e7i)T^{2} \)
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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

−13.44689081995322859996807365927, −12.70357775473733338933092524708, −11.12509822193226668979078130048, −10.36635223599770680586007581064, −8.513604467217673864559819917599, −7.70695546388948028347504651339, −6.50756900086331046388570032666, −4.77442998784181099970788501303, −2.80350696664721591573946160075, −0.898778006579914959275396084852, 2.20951812637372394045156458019, 4.15582262174757537651585554709, 5.28895340928772220712430486856, 7.09692156801365690840091013660, 8.570993233875371754253187206400, 9.574321746096864494424479040726, 10.52308182173312131617828026627, 11.95851514146461260161579899119, 12.83211817998800770977870522565, 14.40791781189625496681617299803

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