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Properties

Label 2-76-4.3-c4-0-24
Degree $2$
Conductor $76$
Sign $0.889 + 0.456i$
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  + (2.08 + 3.41i)2-s − 11.2i·3-s + (−7.30 + 14.2i)4-s + 21.9·5-s + (38.2 − 23.3i)6-s − 62.4i·7-s + (−63.8 + 4.73i)8-s − 44.5·9-s + (45.6 + 74.8i)10-s − 98.2i·11-s + (159. + 81.8i)12-s + 175.·13-s + (213. − 130. i)14-s − 245. i·15-s + (−149. − 208. i)16-s + 328.·17-s + ⋯
L(s)  = 1  + (0.521 + 0.853i)2-s − 1.24i·3-s + (−0.456 + 0.889i)4-s + 0.876·5-s + (1.06 − 0.648i)6-s − 1.27i·7-s + (−0.997 + 0.0740i)8-s − 0.550·9-s + (0.456 + 0.748i)10-s − 0.811i·11-s + (1.10 + 0.568i)12-s + 1.03·13-s + (1.08 − 0.664i)14-s − 1.09i·15-s + (−0.582 − 0.812i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.889 + 0.456i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.25181 - 0.544162i\)
\(L(\frac12)\) \(\approx\) \(2.25181 - 0.544162i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.08 - 3.41i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 + 11.2iT - 81T^{2} \)
5 \( 1 - 21.9T + 625T^{2} \)
7 \( 1 + 62.4iT - 2.40e3T^{2} \)
11 \( 1 + 98.2iT - 1.46e4T^{2} \)
13 \( 1 - 175.T + 2.85e4T^{2} \)
17 \( 1 - 328.T + 8.35e4T^{2} \)
23 \( 1 - 347. iT - 2.79e5T^{2} \)
29 \( 1 + 101.T + 7.07e5T^{2} \)
31 \( 1 - 1.81e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.82e3T + 1.87e6T^{2} \)
41 \( 1 + 1.49e3T + 2.82e6T^{2} \)
43 \( 1 - 83.7iT - 3.41e6T^{2} \)
47 \( 1 - 1.47e3iT - 4.87e6T^{2} \)
53 \( 1 - 5.02e3T + 7.89e6T^{2} \)
59 \( 1 - 2.08e3iT - 1.21e7T^{2} \)
61 \( 1 + 916.T + 1.38e7T^{2} \)
67 \( 1 - 8.17e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.88e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.14e3T + 2.83e7T^{2} \)
79 \( 1 + 2.40e3iT - 3.89e7T^{2} \)
83 \( 1 - 2.22e3iT - 4.74e7T^{2} \)
89 \( 1 + 646.T + 6.27e7T^{2} \)
97 \( 1 - 1.09e4T + 8.85e7T^{2} \)
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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

−13.70407730899948446716803710256, −13.14210918047702152499292142175, −11.90983474628486681861184710873, −10.34242455838671383818001703021, −8.655005083550994948244336322869, −7.50034012330019411731763939589, −6.60989441556897708720270866901, −5.54927811499292776021548105339, −3.52597701868182259753823786179, −1.16964086403946171080812729574, 2.05449023247899764631217673552, 3.67537532972039292475074287767, 5.13834529345865991244616346370, 5.98298435505994490822902854729, 8.775223620077005338237355858092, 9.709986102707947803377774540864, 10.32619523034293115333605550183, 11.63281057053055876977004414958, 12.65493861290356765731635591548, 13.84431694897202977862280873108

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