Properties

Label 2-76-19.14-c4-0-2
Degree $2$
Conductor $76$
Sign $0.928 + 0.372i$
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  + (−7.79 − 1.37i)3-s + (0.638 + 0.535i)5-s + (23.1 + 40.1i)7-s + (−17.1 − 6.25i)9-s + (95.5 − 165. i)11-s + (208. − 36.7i)13-s + (−4.24 − 5.05i)15-s + (504. − 183. i)17-s + (130. + 336. i)19-s + (−125. − 344. i)21-s + (−238. + 200. i)23-s + (−108. − 614. i)25-s + (680. + 393. i)27-s + (67.5 − 185. i)29-s + (−677. + 391. i)31-s + ⋯
L(s)  = 1  + (−0.866 − 0.152i)3-s + (0.0255 + 0.0214i)5-s + (0.472 + 0.818i)7-s + (−0.212 − 0.0771i)9-s + (0.789 − 1.36i)11-s + (1.23 − 0.217i)13-s + (−0.0188 − 0.0224i)15-s + (1.74 − 0.635i)17-s + (0.362 + 0.932i)19-s + (−0.284 − 0.781i)21-s + (−0.450 + 0.378i)23-s + (−0.173 − 0.983i)25-s + (0.934 + 0.539i)27-s + (0.0803 − 0.220i)29-s + (−0.705 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.32749 - 0.256281i\)
\(L(\frac12)\) \(\approx\) \(1.32749 - 0.256281i\)
\(L(3)\) 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 \( 1 + (-130. - 336. i)T \)
good3 \( 1 + (7.79 + 1.37i)T + (76.1 + 27.7i)T^{2} \)
5 \( 1 + (-0.638 - 0.535i)T + (108. + 615. i)T^{2} \)
7 \( 1 + (-23.1 - 40.1i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-95.5 + 165. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-208. + 36.7i)T + (2.68e4 - 9.76e3i)T^{2} \)
17 \( 1 + (-504. + 183. i)T + (6.39e4 - 5.36e4i)T^{2} \)
23 \( 1 + (238. - 200. i)T + (4.85e4 - 2.75e5i)T^{2} \)
29 \( 1 + (-67.5 + 185. i)T + (-5.41e5 - 4.54e5i)T^{2} \)
31 \( 1 + (677. - 391. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.10e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.01e3 - 178. i)T + (2.65e6 + 9.66e5i)T^{2} \)
43 \( 1 + (-867. - 727. i)T + (5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (1.48e3 + 541. i)T + (3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 + (-623. - 742. i)T + (-1.37e6 + 7.77e6i)T^{2} \)
59 \( 1 + (-1.65e3 - 4.55e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (3.86e3 - 3.24e3i)T + (2.40e6 - 1.36e7i)T^{2} \)
67 \( 1 + (70.2 - 192. i)T + (-1.54e7 - 1.29e7i)T^{2} \)
71 \( 1 + (-3.44e3 + 4.10e3i)T + (-4.41e6 - 2.50e7i)T^{2} \)
73 \( 1 + (-1.41e3 + 8.04e3i)T + (-2.66e7 - 9.71e6i)T^{2} \)
79 \( 1 + (-5.64e3 - 995. i)T + (3.66e7 + 1.33e7i)T^{2} \)
83 \( 1 + (2.38e3 + 4.13e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-1.10e4 + 1.94e3i)T + (5.89e7 - 2.14e7i)T^{2} \)
97 \( 1 + (-613. - 1.68e3i)T + (-6.78e7 + 5.69e7i)T^{2} \)
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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

−13.84246762084509965961783450664, −12.17038795972245154976691061534, −11.68904900056284238618824324441, −10.64968024798474271484588313675, −9.054474548958259036059572270041, −7.995034428941278685004491855207, −6.04371930458049287806126178442, −5.62095639196438415683661166973, −3.43562835380515291680564434257, −1.01805101086816540227444021817, 1.26031726551423977762649939881, 3.92484461329498489741489043521, 5.28073933002700703265852554646, 6.63022418237021035841818953926, 7.916210268474413239999464281291, 9.514079133632512755356996955043, 10.69179799043532483311705187843, 11.51754881607767398050340300969, 12.56471888940673237506274329607, 13.91627593828646106229576606722

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