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Properties

Label 2-76-76.11-c2-0-7
Degree $2$
Conductor $76$
Sign $0.625 - 0.780i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 1.17i)2-s + (0.851 + 0.491i)3-s + (1.23 + 3.80i)4-s + (1.71 − 2.97i)5-s + (0.799 + 1.79i)6-s + 2.43i·7-s + (−2.47 + 7.60i)8-s + (−4.01 − 6.95i)9-s + (6.27 − 2.79i)10-s + 7.76i·11-s + (−0.819 + 3.84i)12-s + (−4.63 − 8.02i)13-s + (−2.86 + 3.94i)14-s + (2.92 − 1.68i)15-s + (−12.9 + 9.39i)16-s + (8.87 − 15.3i)17-s + ⋯
L(s)  = 1  + (0.808 + 0.587i)2-s + (0.283 + 0.163i)3-s + (0.308 + 0.951i)4-s + (0.343 − 0.594i)5-s + (0.133 + 0.299i)6-s + 0.348i·7-s + (−0.309 + 0.950i)8-s + (−0.446 − 0.772i)9-s + (0.627 − 0.279i)10-s + 0.705i·11-s + (−0.0682 + 0.320i)12-s + (−0.356 − 0.617i)13-s + (−0.204 + 0.281i)14-s + (0.194 − 0.112i)15-s + (−0.809 + 0.587i)16-s + (0.521 − 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.625 - 0.780i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81006 + 0.869169i\)
\(L(\frac12)\) \(\approx\) \(1.81006 + 0.869169i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 1.17i)T \)
19 \( 1 + (6.84 + 17.7i)T \)
good3 \( 1 + (-0.851 - 0.491i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.71 + 2.97i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 - 7.76iT - 121T^{2} \)
13 \( 1 + (4.63 + 8.02i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-8.87 + 15.3i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (10.0 - 5.80i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.64 - 4.57i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 10.2iT - 961T^{2} \)
37 \( 1 + 7.71T + 1.36e3T^{2} \)
41 \( 1 + (25.2 - 43.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-43.2 - 24.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (74.9 - 43.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-17.3 - 30.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-63.9 - 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-40.4 - 70.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-14.8 + 8.60i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (47.9 + 27.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-27.9 + 48.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (95.2 + 54.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 31.3iT - 6.88e3T^{2} \)
89 \( 1 + (5.29 + 9.17i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-65.4 + 113. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.58428116004886830628216329901, −13.37019962234529091951994092216, −12.48295838586246802779885291171, −11.55561083388682428445050828619, −9.652380053019352882539942933407, −8.641577224741703631983327202640, −7.27496024463053613408601243467, −5.82659968499117720335083634523, −4.67223412409002291819520726277, −2.90097562669885987289895631028, 2.16897626921815656437181325946, 3.74106094589641158799850424596, 5.50094444637837319627823107760, 6.73478008031704094100224227428, 8.365410663509295046713497211966, 10.09827450785851762007276667287, 10.79338452729952514181778123030, 11.98999178319211073414667723204, 13.17774868589021991711706372303, 14.23396173130384947180052746974

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