Properties

Label 2-76-76.75-c5-0-40
Degree $2$
Conductor $76$
Sign $0.703 + 0.711i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.30 − 1.95i)2-s + 7.37·3-s + (24.3 − 20.7i)4-s + 74.1·5-s + (39.1 − 14.4i)6-s − 80.5i·7-s + (88.5 − 157. i)8-s − 188.·9-s + (393. − 145. i)10-s + 299. i·11-s + (179. − 153. i)12-s + 472. i·13-s + (−157. − 427. i)14-s + 546.·15-s + (161. − 1.01e3i)16-s + 861.·17-s + ⋯
L(s)  = 1  + (0.938 − 0.345i)2-s + 0.473·3-s + (0.760 − 0.648i)4-s + 1.32·5-s + (0.443 − 0.163i)6-s − 0.621i·7-s + (0.489 − 0.872i)8-s − 0.776·9-s + (1.24 − 0.458i)10-s + 0.747i·11-s + (0.359 − 0.306i)12-s + 0.775i·13-s + (−0.214 − 0.582i)14-s + 0.627·15-s + (0.157 − 0.987i)16-s + 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.703 + 0.711i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.703 + 0.711i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.83601 - 1.60194i\)
\(L(\frac12)\) \(\approx\) \(3.83601 - 1.60194i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.30 + 1.95i)T \)
19 \( 1 + (115. + 1.56e3i)T \)
good3 \( 1 - 7.37T + 243T^{2} \)
5 \( 1 - 74.1T + 3.12e3T^{2} \)
7 \( 1 + 80.5iT - 1.68e4T^{2} \)
11 \( 1 - 299. iT - 1.61e5T^{2} \)
13 \( 1 - 472. iT - 3.71e5T^{2} \)
17 \( 1 - 861.T + 1.41e6T^{2} \)
23 \( 1 + 840. iT - 6.43e6T^{2} \)
29 \( 1 - 5.15e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.42e3T + 2.86e7T^{2} \)
37 \( 1 - 4.13e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.96e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.19e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.58e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.88e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.14e4T + 7.14e8T^{2} \)
61 \( 1 + 2.94e4T + 8.44e8T^{2} \)
67 \( 1 - 2.17e4T + 1.35e9T^{2} \)
71 \( 1 + 2.65e3T + 1.80e9T^{2} \)
73 \( 1 + 4.62e4T + 2.07e9T^{2} \)
79 \( 1 + 7.59e4T + 3.07e9T^{2} \)
83 \( 1 + 3.50e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.13e4iT - 5.58e9T^{2} \)
97 \( 1 + 9.09e4iT - 8.58e9T^{2} \)
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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

−13.61576505311098336242137229002, −12.58448632080635519409000315398, −11.23706691939028291353700502529, −10.11215611182624121262298122179, −9.152558217224359862942184039295, −7.18895327758551622170985797237, −5.98749936501980348241669949561, −4.68545010686385108992475975638, −2.94796388837761080353070184874, −1.64838048834386511948367307271, 2.15634283565550760657748695869, 3.35613474819877572554461912246, 5.62537779099939749243101795079, 5.88492897268504899340611813969, 7.83052647808643748382531243321, 8.956825892096405113443294756802, 10.37815034203721224737554924521, 11.75242002818186154803430531309, 12.89656926503136734947535153820, 13.86060003627633789409735327889

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