Properties

Label 2-76-76.75-c5-0-12
Degree $2$
Conductor $76$
Sign $-0.0728 - 0.997i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0728 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0728 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0728 - 0.997i$
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: \((2,\ 76,\ (\ :5/2),\ -0.0728 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.10767 + 1.19156i\)
\(L(\frac12)\) \(\approx\) \(1.10767 + 1.19156i\)
\(L(\frac{7}{2})\) 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

−13.74503402738647040169802724254, −12.26393787736355265312835450616, −11.96921953644022066331145356596, −10.99104135081069288547637798209, −9.585149195199398028811719340686, −7.62684249556390632048646548367, −6.28789445408076248952360788076, −5.33261951572008441992497608333, −4.03681784886748282966161692365, −2.01322954597781533771666534228, 0.56251850417799994245517987494, 3.19930617558687408546463562235, 4.67967394504300711813959400331, 5.85761372930443328387696541714, 6.93392305433389430704987418051, 8.301344674319482575443320895351, 10.64700981668108041537711787934, 11.06671346336774183193845985059, 12.16833080701488144926680438447, 13.21410096049412949076965452144

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