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Properties

Label 2-76-76.75-c5-0-37
Degree $2$
Conductor $76$
Sign $-0.981 - 0.191i$
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  + (−3.92 − 4.07i)2-s + 5.43·3-s + (−1.17 + 31.9i)4-s + 4.82·5-s + (−21.3 − 22.1i)6-s + 68.2i·7-s + (134. − 120. i)8-s − 213.·9-s + (−18.9 − 19.6i)10-s − 223. i·11-s + (−6.40 + 173. i)12-s − 627. i·13-s + (278. − 267. i)14-s + 26.2·15-s + (−1.02e3 − 75.3i)16-s − 1.56e3·17-s + ⋯
L(s)  = 1  + (−0.693 − 0.720i)2-s + 0.348·3-s + (−0.0368 + 0.999i)4-s + 0.0862·5-s + (−0.242 − 0.251i)6-s + 0.526i·7-s + (0.745 − 0.666i)8-s − 0.878·9-s + (−0.0598 − 0.0621i)10-s − 0.555i·11-s + (−0.0128 + 0.348i)12-s − 1.03i·13-s + (0.379 − 0.365i)14-s + 0.0300·15-s + (−0.997 − 0.0736i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.981 - 0.191i$
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: \((2,\ 76,\ (\ :5/2),\ -0.981 - 0.191i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0278841 + 0.289238i\)
\(L(\frac12)\) \(\approx\) \(0.0278841 + 0.289238i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.92 + 4.07i)T \)
19 \( 1 + (-243. + 1.55e3i)T \)
good3 \( 1 - 5.43T + 243T^{2} \)
5 \( 1 - 4.82T + 3.12e3T^{2} \)
7 \( 1 - 68.2iT - 1.68e4T^{2} \)
11 \( 1 + 223. iT - 1.61e5T^{2} \)
13 \( 1 + 627. iT - 3.71e5T^{2} \)
17 \( 1 + 1.56e3T + 1.41e6T^{2} \)
23 \( 1 - 3.50e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.77e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.30e3T + 2.86e7T^{2} \)
37 \( 1 + 9.40e3iT - 6.93e7T^{2} \)
41 \( 1 + 8.20e3iT - 1.15e8T^{2} \)
43 \( 1 + 68.8iT - 1.47e8T^{2} \)
47 \( 1 - 1.17e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.74e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.96e3T + 7.14e8T^{2} \)
61 \( 1 - 3.00e4T + 8.44e8T^{2} \)
67 \( 1 + 2.23e4T + 1.35e9T^{2} \)
71 \( 1 + 4.46e4T + 1.80e9T^{2} \)
73 \( 1 + 5.85e4T + 2.07e9T^{2} \)
79 \( 1 - 7.58e4T + 3.07e9T^{2} \)
83 \( 1 - 2.00e3iT - 3.93e9T^{2} \)
89 \( 1 - 1.16e5iT - 5.58e9T^{2} \)
97 \( 1 + 7.84e4iT - 8.58e9T^{2} \)
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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

−12.90223923275014592663131533886, −11.54504128531114655042276496135, −10.87013231954636680709275889241, −9.321140587011011992138006795415, −8.674101993527048416092302345829, −7.44697941941538853584046437595, −5.58466722136983129666699455570, −3.49102881990410274933234144660, −2.24861843840115966036635667884, −0.14057076151637461071896649923, 1.99326394649982503218460286124, 4.37332335631441628139574115876, 6.05129418925107055863110450618, 7.20320448649367219501162275725, 8.429653147545789831708064303119, 9.357655440731246856806326562635, 10.53157443866556826867869082979, 11.68217937747986199317269409696, 13.43414666445232184879736885719, 14.30849722256007297431987955039

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