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Properties

Label 2-76-4.3-c4-0-9
Degree $2$
Conductor $76$
Sign $0.968 + 0.247i$
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  + (−2.45 − 3.15i)2-s + 2.52i·3-s + (−3.96 + 15.5i)4-s − 40.2·5-s + (7.98 − 6.19i)6-s − 43.5i·7-s + (58.6 − 25.5i)8-s + 74.6·9-s + (98.8 + 127. i)10-s + 155. i·11-s + (−39.1 − 10.0i)12-s + 209.·13-s + (−137. + 106. i)14-s − 101. i·15-s + (−224. − 122. i)16-s + 408.·17-s + ⋯
L(s)  = 1  + (−0.613 − 0.789i)2-s + 0.280i·3-s + (−0.247 + 0.968i)4-s − 1.61·5-s + (0.221 − 0.172i)6-s − 0.889i·7-s + (0.917 − 0.398i)8-s + 0.921·9-s + (0.988 + 1.27i)10-s + 1.28i·11-s + (−0.272 − 0.0695i)12-s + 1.24·13-s + (−0.702 + 0.545i)14-s − 0.452i·15-s + (−0.877 − 0.480i)16-s + 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.968 + 0.247i$
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: \((2,\ 76,\ (\ :2),\ 0.968 + 0.247i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.944310 - 0.118831i\)
\(L(\frac12)\) \(\approx\) \(0.944310 - 0.118831i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.45 + 3.15i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 - 2.52iT - 81T^{2} \)
5 \( 1 + 40.2T + 625T^{2} \)
7 \( 1 + 43.5iT - 2.40e3T^{2} \)
11 \( 1 - 155. iT - 1.46e4T^{2} \)
13 \( 1 - 209.T + 2.85e4T^{2} \)
17 \( 1 - 408.T + 8.35e4T^{2} \)
23 \( 1 - 518. iT - 2.79e5T^{2} \)
29 \( 1 + 466.T + 7.07e5T^{2} \)
31 \( 1 + 1.45e3iT - 9.23e5T^{2} \)
37 \( 1 + 369.T + 1.87e6T^{2} \)
41 \( 1 - 1.76e3T + 2.82e6T^{2} \)
43 \( 1 - 1.43e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.45e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.39e3T + 7.89e6T^{2} \)
59 \( 1 - 6.07e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.20e3T + 1.38e7T^{2} \)
67 \( 1 + 4.52e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.59e3iT - 2.54e7T^{2} \)
73 \( 1 + 567.T + 2.83e7T^{2} \)
79 \( 1 - 4.90e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.83e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.65e3T + 6.27e7T^{2} \)
97 \( 1 - 234.T + 8.85e7T^{2} \)
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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

−13.34803887695254874017746872234, −12.37475816022372945696664671394, −11.40484816311111418548656602828, −10.46146084092789233166531252592, −9.421770066420781144803423605909, −7.76012481789317471064421069817, −7.37669659407796359467722771416, −4.30116044503510896305095350718, −3.68012410309349164527531006137, −1.07237142658069582415230449813, 0.859802479887120544135245613597, 3.75842950645098603431465046539, 5.57876154505244446088213579881, 6.94930038993688761936782077082, 8.155947948258757816648051368808, 8.705603038002511429416610138294, 10.45669088641402548433415446823, 11.54000805245149485992056375858, 12.64572293798165612169178936901, 14.10594008493707803599865290448

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