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Properties

Label 2-76-76.75-c5-0-41
Degree $2$
Conductor $76$
Sign $-0.0528 + 0.998i$
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  + (3.09 + 4.73i)2-s + 0.774·3-s + (−12.8 + 29.3i)4-s − 32.1·5-s + (2.39 + 3.66i)6-s − 237. i·7-s + (−178. + 29.7i)8-s − 242.·9-s + (−99.3 − 152. i)10-s + 381. i·11-s + (−9.96 + 22.6i)12-s − 905. i·13-s + (1.12e3 − 734. i)14-s − 24.8·15-s + (−693. − 753. i)16-s + 1.01e3·17-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + 0.0496·3-s + (−0.401 + 0.915i)4-s − 0.574·5-s + (0.0271 + 0.0415i)6-s − 1.83i·7-s + (−0.986 + 0.164i)8-s − 0.997·9-s + (−0.314 − 0.480i)10-s + 0.951i·11-s + (−0.0199 + 0.0454i)12-s − 1.48i·13-s + (1.53 − 1.00i)14-s − 0.0285·15-s + (−0.676 − 0.736i)16-s + 0.852·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0528 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0528 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0528 + 0.998i$
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: \((2,\ 76,\ (\ :5/2),\ -0.0528 + 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.420464 - 0.443324i\)
\(L(\frac12)\) \(\approx\) \(0.420464 - 0.443324i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.09 - 4.73i)T \)
19 \( 1 + (1.47e3 - 555. i)T \)
good3 \( 1 - 0.774T + 243T^{2} \)
5 \( 1 + 32.1T + 3.12e3T^{2} \)
7 \( 1 + 237. iT - 1.68e4T^{2} \)
11 \( 1 - 381. iT - 1.61e5T^{2} \)
13 \( 1 + 905. iT - 3.71e5T^{2} \)
17 \( 1 - 1.01e3T + 1.41e6T^{2} \)
23 \( 1 - 724. iT - 6.43e6T^{2} \)
29 \( 1 + 6.04e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.45e3T + 2.86e7T^{2} \)
37 \( 1 + 8.56e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.73e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.12e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.08e4iT - 2.29e8T^{2} \)
53 \( 1 + 7.04e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.62e4T + 7.14e8T^{2} \)
61 \( 1 + 4.54e4T + 8.44e8T^{2} \)
67 \( 1 - 4.35e4T + 1.35e9T^{2} \)
71 \( 1 + 5.61e4T + 1.80e9T^{2} \)
73 \( 1 - 1.13e4T + 2.07e9T^{2} \)
79 \( 1 - 9.07e4T + 3.07e9T^{2} \)
83 \( 1 + 6.32e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.33e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.00e4iT - 8.58e9T^{2} \)
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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

−13.39862504897950453219419751708, −12.47968636762836722818295956974, −11.12636295587049050466114452520, −9.858119722359982846678670488443, −7.937055490464407083233120999500, −7.61154848767724124635480274622, −6.03938318339556026774320115783, −4.48831124284324468945061469161, −3.37844651278664719677890418243, −0.20957895668840585493079912578, 2.17282820419851205017853405548, 3.46496203803693227246429104343, 5.22322007785155151178559321831, 6.22696765941339107948518642016, 8.614803768493328606302219882949, 9.102711109807715979394840401474, 10.92477701544025262964868250800, 11.81565750029046997765061665303, 12.30167836776221793390925722912, 13.84420769372627581198871293852

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