Properties

Label 2-76-19.3-c4-0-2
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: $\chi_{76} (41, \cdot )$
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

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

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