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Properties

Label 2-76-76.75-c5-0-11
Degree $2$
Conductor $76$
Sign $-0.390 - 0.920i$
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  + (−5.61 + 0.669i)2-s + 15.7·3-s + (31.1 − 7.52i)4-s − 23.2·5-s + (−88.2 + 10.5i)6-s + 110. i·7-s + (−169. + 63.0i)8-s + 4.08·9-s + (130. − 15.5i)10-s + 468. i·11-s + (488. − 118. i)12-s − 546. i·13-s + (−73.6 − 618. i)14-s − 365.·15-s + (910. − 467. i)16-s − 170.·17-s + ⋯
L(s)  = 1  + (−0.992 + 0.118i)2-s + 1.00·3-s + (0.971 − 0.235i)4-s − 0.416·5-s + (−1.00 + 0.119i)6-s + 0.848i·7-s + (−0.937 + 0.348i)8-s + 0.0168·9-s + (0.413 − 0.0492i)10-s + 1.16i·11-s + (0.980 − 0.237i)12-s − 0.897i·13-s + (−0.100 − 0.842i)14-s − 0.419·15-s + (0.889 − 0.456i)16-s − 0.143·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.390 - 0.920i$
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: \((2,\ 76,\ (\ :5/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.567974 + 0.858312i\)
\(L(\frac12)\) \(\approx\) \(0.567974 + 0.858312i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.61 - 0.669i)T \)
19 \( 1 + (-257. - 1.55e3i)T \)
good3 \( 1 - 15.7T + 243T^{2} \)
5 \( 1 + 23.2T + 3.12e3T^{2} \)
7 \( 1 - 110. iT - 1.68e4T^{2} \)
11 \( 1 - 468. iT - 1.61e5T^{2} \)
13 \( 1 + 546. iT - 3.71e5T^{2} \)
17 \( 1 + 170.T + 1.41e6T^{2} \)
23 \( 1 - 3.43e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.80e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.39e3T + 2.86e7T^{2} \)
37 \( 1 - 6.13e3iT - 6.93e7T^{2} \)
41 \( 1 + 3.81e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.17e4iT - 2.29e8T^{2} \)
53 \( 1 - 867. iT - 4.18e8T^{2} \)
59 \( 1 - 4.51e3T + 7.14e8T^{2} \)
61 \( 1 + 2.80e4T + 8.44e8T^{2} \)
67 \( 1 - 3.94e4T + 1.35e9T^{2} \)
71 \( 1 + 2.08e4T + 1.80e9T^{2} \)
73 \( 1 - 8.65e4T + 2.07e9T^{2} \)
79 \( 1 + 4.65e4T + 3.07e9T^{2} \)
83 \( 1 + 1.01e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.02e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.55e5iT - 8.58e9T^{2} \)
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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

−14.19785553116412885900971524499, −12.56643725451333272429745893677, −11.63318427182488835157400871828, −10.14379615855169034792274793479, −9.200132666719341942716157576172, −8.201386452325273242033344424950, −7.39276282398560343037414567444, −5.63663567971673375913583052833, −3.26280524134630864972994772562, −1.90009900670493019460665633627, 0.52067297435926457271510227749, 2.50681421846245138473728403765, 3.88452113603511783908640470344, 6.45833989289458569495664792152, 7.74840993184807372161725569012, 8.568106408705311878024709251756, 9.536286182051517155015441908268, 10.89109694894925748963710086480, 11.71096084429073656744293615497, 13.39419515416217212660138668102

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