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Properties

Label 2-76-4.3-c4-0-23
Degree $2$
Conductor $76$
Sign $0.255 + 0.966i$
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  + (0.515 + 3.96i)2-s + 1.66i·3-s + (−15.4 + 4.08i)4-s − 27.3·5-s + (−6.59 + 0.856i)6-s − 39.8i·7-s + (−24.1 − 59.2i)8-s + 78.2·9-s + (−14.1 − 108. i)10-s − 69.8i·11-s + (−6.79 − 25.7i)12-s − 87.6·13-s + (157. − 20.5i)14-s − 45.4i·15-s + (222. − 126. i)16-s − 440.·17-s + ⋯
L(s)  = 1  + (0.128 + 0.991i)2-s + 0.184i·3-s + (−0.966 + 0.255i)4-s − 1.09·5-s + (−0.183 + 0.0237i)6-s − 0.812i·7-s + (−0.377 − 0.925i)8-s + 0.965·9-s + (−0.141 − 1.08i)10-s − 0.576i·11-s + (−0.0471 − 0.178i)12-s − 0.518·13-s + (0.805 − 0.104i)14-s − 0.202i·15-s + (0.869 − 0.494i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.255 + 0.966i$
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: \((2,\ 76,\ (\ :2),\ 0.255 + 0.966i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.360464 - 0.277581i\)
\(L(\frac12)\) \(\approx\) \(0.360464 - 0.277581i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.515 - 3.96i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 - 1.66iT - 81T^{2} \)
5 \( 1 + 27.3T + 625T^{2} \)
7 \( 1 + 39.8iT - 2.40e3T^{2} \)
11 \( 1 + 69.8iT - 1.46e4T^{2} \)
13 \( 1 + 87.6T + 2.85e4T^{2} \)
17 \( 1 + 440.T + 8.35e4T^{2} \)
23 \( 1 + 429. iT - 2.79e5T^{2} \)
29 \( 1 + 405.T + 7.07e5T^{2} \)
31 \( 1 + 942. iT - 9.23e5T^{2} \)
37 \( 1 + 1.54e3T + 1.87e6T^{2} \)
41 \( 1 + 2.00e3T + 2.82e6T^{2} \)
43 \( 1 - 3.32e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.86e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.07e3T + 7.89e6T^{2} \)
59 \( 1 - 72.9iT - 1.21e7T^{2} \)
61 \( 1 - 4.60e3T + 1.38e7T^{2} \)
67 \( 1 - 1.19e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.79e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.10e3T + 2.83e7T^{2} \)
79 \( 1 + 4.87e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.15e4iT - 4.74e7T^{2} \)
89 \( 1 + 7.65e3T + 6.27e7T^{2} \)
97 \( 1 - 180.T + 8.85e7T^{2} \)
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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

−13.61903684923472991721659249521, −12.77078322839742695639699486117, −11.34766692284452652567747150100, −10.03856789666599118507807655519, −8.646329980069801932940639957966, −7.52485055972241252093288352851, −6.67376041962081519868172663111, −4.69363759653589398640275237829, −3.84254668583147165587772884224, −0.22373837167550656318749459084, 1.98799654734776004535046399460, 3.80131763242171021765407527225, 5.01058655477724500946604379606, 7.08445751368843629790784687926, 8.470325130158411519386331248926, 9.613754354727446417711335727508, 10.87932766791717755424276598363, 12.00728372141719682029704741855, 12.51287878965151025546710371034, 13.69219536187446524895048878605

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