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Properties

Label 2-76-4.3-c6-0-31
Degree $2$
Conductor $76$
Sign $0.943 + 0.332i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.52 + 4.62i)2-s − 11.8i·3-s + (21.2 − 60.3i)4-s + 179.·5-s + (54.7 + 77.3i)6-s − 39.4i·7-s + (140. + 492. i)8-s + 588.·9-s + (−1.17e3 + 830. i)10-s − 536. i·11-s + (−715. − 251. i)12-s − 1.50e3·13-s + (182. + 257. i)14-s − 2.13e3i·15-s + (−3.19e3 − 2.56e3i)16-s + 2.14e3·17-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)2-s − 0.438i·3-s + (0.332 − 0.943i)4-s + 1.43·5-s + (0.253 + 0.358i)6-s − 0.115i·7-s + (0.273 + 0.961i)8-s + 0.807·9-s + (−1.17 + 0.830i)10-s − 0.403i·11-s + (−0.414 − 0.145i)12-s − 0.686·13-s + (0.0664 + 0.0939i)14-s − 0.631i·15-s + (−0.779 − 0.626i)16-s + 0.436·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ 0.943 + 0.332i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.68098 - 0.287371i\)
\(L(\frac12)\) \(\approx\) \(1.68098 - 0.287371i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.52 - 4.62i)T \)
19 \( 1 + 1.57e3iT \)
good3 \( 1 + 11.8iT - 729T^{2} \)
5 \( 1 - 179.T + 1.56e4T^{2} \)
7 \( 1 + 39.4iT - 1.17e5T^{2} \)
11 \( 1 + 536. iT - 1.77e6T^{2} \)
13 \( 1 + 1.50e3T + 4.82e6T^{2} \)
17 \( 1 - 2.14e3T + 2.41e7T^{2} \)
23 \( 1 - 261. iT - 1.48e8T^{2} \)
29 \( 1 - 4.05e4T + 5.94e8T^{2} \)
31 \( 1 - 2.64e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.75e4T + 2.56e9T^{2} \)
41 \( 1 - 3.06e4T + 4.75e9T^{2} \)
43 \( 1 + 1.31e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.98e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.28e5T + 2.21e10T^{2} \)
59 \( 1 - 1.88e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.66e5T + 5.15e10T^{2} \)
67 \( 1 - 2.67e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.63e5T + 1.51e11T^{2} \)
79 \( 1 + 4.86e4iT - 2.43e11T^{2} \)
83 \( 1 - 3.17e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.02e4T + 4.96e11T^{2} \)
97 \( 1 - 6.66e5T + 8.32e11T^{2} \)
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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

−13.55026533347152246057114801774, −12.19696826818615421530720466189, −10.46714564274706439083942005196, −9.868481983770175898790439512252, −8.704055341884997206145168828566, −7.27667232546518241721873104723, −6.33365418930896895017434837700, −5.12751024897845252821281889743, −2.24546937235672305092697036940, −0.985939940966534913858702494932, 1.35428725117179816475926405623, 2.65611172047539342893377116739, 4.58122202180303281987179781111, 6.33061486124127583844000475444, 7.74078750340546459577290530749, 9.395724624679533367774996640031, 9.814676303082257319085238001904, 10.71851993567963676072990322272, 12.25770273774907536285461213204, 13.09721568078147299630183562457

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