Properties

Label 2-76-19.2-c4-0-6
Degree $2$
Conductor $76$
Sign $-0.907 + 0.420i$
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  + (10.9 − 13.1i)3-s + (−41.8 + 15.2i)5-s + (−26.2 − 45.5i)7-s + (−36.7 − 208. i)9-s + (−55.1 + 95.5i)11-s + (31.8 + 37.9i)13-s + (−260. + 716. i)15-s + (62.4 − 354. i)17-s + (346. − 100. i)19-s + (−885. − 156. i)21-s + (−253. − 92.2i)23-s + (1.04e3 − 875. i)25-s + (−1.93e3 − 1.11e3i)27-s + (−563. + 99.3i)29-s + (910. − 525. i)31-s + ⋯
L(s)  = 1  + (1.22 − 1.45i)3-s + (−1.67 + 0.609i)5-s + (−0.536 − 0.928i)7-s + (−0.454 − 2.57i)9-s + (−0.455 + 0.789i)11-s + (0.188 + 0.224i)13-s + (−1.15 + 3.18i)15-s + (0.216 − 1.22i)17-s + (0.960 − 0.278i)19-s + (−2.00 − 0.354i)21-s + (−0.479 − 0.174i)23-s + (1.67 − 1.40i)25-s + (−2.65 − 1.53i)27-s + (−0.670 + 0.118i)29-s + (0.947 − 0.546i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.907 + 0.420i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.274683 - 1.24684i\)
\(L(\frac12)\) \(\approx\) \(0.274683 - 1.24684i\)
\(L(3)\) 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 \( 1 + (-346. + 100. i)T \)
good3 \( 1 + (-10.9 + 13.1i)T + (-14.0 - 79.7i)T^{2} \)
5 \( 1 + (41.8 - 15.2i)T + (478. - 401. i)T^{2} \)
7 \( 1 + (26.2 + 45.5i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (55.1 - 95.5i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-31.8 - 37.9i)T + (-4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-62.4 + 354. i)T + (-7.84e4 - 2.85e4i)T^{2} \)
23 \( 1 + (253. + 92.2i)T + (2.14e5 + 1.79e5i)T^{2} \)
29 \( 1 + (563. - 99.3i)T + (6.64e5 - 2.41e5i)T^{2} \)
31 \( 1 + (-910. + 525. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 527. iT - 1.87e6T^{2} \)
41 \( 1 + (-407. + 485. i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (910. - 331. i)T + (2.61e6 - 2.19e6i)T^{2} \)
47 \( 1 + (22.9 + 130. i)T + (-4.58e6 + 1.66e6i)T^{2} \)
53 \( 1 + (203. - 558. i)T + (-6.04e6 - 5.07e6i)T^{2} \)
59 \( 1 + (-3.75e3 - 661. i)T + (1.13e7 + 4.14e6i)T^{2} \)
61 \( 1 + (-1.59e3 - 579. i)T + (1.06e7 + 8.89e6i)T^{2} \)
67 \( 1 + (-2.49e3 + 439. i)T + (1.89e7 - 6.89e6i)T^{2} \)
71 \( 1 + (-1.82e3 - 5.02e3i)T + (-1.94e7 + 1.63e7i)T^{2} \)
73 \( 1 + (-500. - 419. i)T + (4.93e6 + 2.79e7i)T^{2} \)
79 \( 1 + (53.2 - 63.4i)T + (-6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (3.32e3 + 5.76e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-3.75e3 - 4.47e3i)T + (-1.08e7 + 6.17e7i)T^{2} \)
97 \( 1 + (1.42e4 + 2.50e3i)T + (8.31e7 + 3.02e7i)T^{2} \)
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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

−13.38878851590591423536143639102, −12.31004085278450161366971804698, −11.46720406936709270609475702389, −9.717241586111556398804162336640, −8.201168292422202187284544492775, −7.31351120122376137678790815871, −6.97905395342475437858157649020, −3.89901175396619903667826060180, −2.79434061490788684389114294930, −0.56819780491625099908756089615, 3.12080936508121241949160742968, 3.92979211386820825047526213536, 5.33647417514499620562513131876, 8.050219036601814330113364256103, 8.410630484739021715139885334327, 9.536243850471069217059952663189, 10.79204165016851341464400036299, 11.97134761175377282653214772922, 13.24793145424029980221934685147, 14.62275011930843334476801184088

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