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Properties

Label 2-76-76.11-c2-0-13
Degree $2$
Conductor $76$
Sign $0.839 + 0.543i$
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  + (1.19 − 1.60i)2-s + (3.65 + 2.11i)3-s + (−1.13 − 3.83i)4-s + (−1.06 + 1.84i)5-s + (7.76 − 3.33i)6-s − 1.82i·7-s + (−7.50 − 2.77i)8-s + (4.42 + 7.65i)9-s + (1.68 + 3.92i)10-s + 13.8i·11-s + (3.94 − 16.4i)12-s + (−9.95 − 17.2i)13-s + (−2.92 − 2.18i)14-s + (−7.80 + 4.50i)15-s + (−13.4 + 8.70i)16-s + (3.83 − 6.63i)17-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)2-s + (1.21 + 0.703i)3-s + (−0.283 − 0.958i)4-s + (−0.213 + 0.369i)5-s + (1.29 − 0.555i)6-s − 0.260i·7-s + (−0.938 − 0.346i)8-s + (0.491 + 0.850i)9-s + (0.168 + 0.392i)10-s + 1.25i·11-s + (0.329 − 1.36i)12-s + (−0.765 − 1.32i)13-s + (−0.208 − 0.155i)14-s + (−0.520 + 0.300i)15-s + (−0.839 + 0.544i)16-s + (0.225 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.839 + 0.543i$
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: \((2,\ 76,\ (\ :1),\ 0.839 + 0.543i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.98775 - 0.586735i\)
\(L(\frac12)\) \(\approx\) \(1.98775 - 0.586735i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 1.60i)T \)
19 \( 1 + (16.7 - 9.04i)T \)
good3 \( 1 + (-3.65 - 2.11i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.06 - 1.84i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 1.82iT - 49T^{2} \)
11 \( 1 - 13.8iT - 121T^{2} \)
13 \( 1 + (9.95 + 17.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-3.83 + 6.63i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-9.14 + 5.28i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.0147 + 0.0255i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 42.2iT - 961T^{2} \)
37 \( 1 - 19.5T + 1.36e3T^{2} \)
41 \( 1 + (-15.7 + 27.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (51.3 + 29.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-77.5 + 44.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (29.8 + 51.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-63.9 - 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.6 + 40.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.77 + 3.33i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (24.1 + 13.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (46.2 - 80.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (110. + 63.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 88.9iT - 6.88e3T^{2} \)
89 \( 1 + (-31.2 - 54.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-64.8 + 112. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.37662009836366468044740669015, −13.10102513820790792736850884871, −12.17547392715344072887098075148, −10.50787711109467428760463454820, −10.00039508081952118220000768965, −8.765127578970008095593793945650, −7.23183594104162332434452870692, −5.05037914095214243653575797166, −3.74231640506929053825979560609, −2.54022420355908737530365696298, 2.68727327236097734765281139640, 4.32149239307992044317100947572, 6.16956360888017259078840951873, 7.44526216432246338167325896651, 8.470792803584949406554406668570, 9.144048918980787928313253085970, 11.50310350928035119104421131193, 12.66540490873473168826836931819, 13.49572582737688566387492033298, 14.32020634561947216450648667130

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