Properties

Label 2-76-76.11-c2-0-6
Degree $2$
Conductor $76$
Sign $-0.0843 - 0.996i$
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  + (0.545 + 1.92i)2-s + (3.88 + 2.24i)3-s + (−3.40 + 2.09i)4-s + (−0.133 + 0.231i)5-s + (−2.19 + 8.70i)6-s − 7.24i·7-s + (−5.89 − 5.40i)8-s + (5.56 + 9.64i)9-s + (−0.518 − 0.130i)10-s − 11.7i·11-s + (−17.9 + 0.519i)12-s + (4.17 + 7.22i)13-s + (13.9 − 3.95i)14-s + (−1.03 + 0.600i)15-s + (7.18 − 14.2i)16-s + (−7.11 + 12.3i)17-s + ⋯
L(s)  = 1  + (0.272 + 0.962i)2-s + (1.29 + 0.747i)3-s + (−0.851 + 0.524i)4-s + (−0.0267 + 0.0463i)5-s + (−0.366 + 1.45i)6-s − 1.03i·7-s + (−0.737 − 0.675i)8-s + (0.618 + 1.07i)9-s + (−0.0518 − 0.0130i)10-s − 1.06i·11-s + (−1.49 + 0.0433i)12-s + (0.320 + 0.555i)13-s + (0.996 − 0.282i)14-s + (−0.0693 + 0.0400i)15-s + (0.449 − 0.893i)16-s + (−0.418 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0843 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0843 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0843 - 0.996i$
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: \((2,\ 76,\ (\ :1),\ -0.0843 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.23330 + 1.34214i\)
\(L(\frac12)\) \(\approx\) \(1.23330 + 1.34214i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.545 - 1.92i)T \)
19 \( 1 + (11.9 - 14.7i)T \)
good3 \( 1 + (-3.88 - 2.24i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (0.133 - 0.231i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 7.24iT - 49T^{2} \)
11 \( 1 + 11.7iT - 121T^{2} \)
13 \( 1 + (-4.17 - 7.22i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (7.11 - 12.3i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-21.4 + 12.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (25.7 + 44.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 26.3iT - 961T^{2} \)
37 \( 1 + 20.1T + 1.36e3T^{2} \)
41 \( 1 + (38.5 - 66.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.0 - 17.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (31.9 - 18.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (8.75 + 15.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (82.8 + 47.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-26.2 - 45.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (11.8 - 6.86i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (39.7 + 22.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-20.0 + 34.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-106. - 61.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 108. iT - 6.88e3T^{2} \)
89 \( 1 + (26.2 + 45.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-29.9 + 51.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.65743788968590825463526240359, −13.72791740166631763653323994352, −13.11234958920264904441667487657, −11.01999803336501113395198126558, −9.685851268514759748087500670653, −8.650256729171675958194698056696, −7.82010378758799671488570804180, −6.31724588793799666508238805748, −4.36209441630565040641674475564, −3.45928122962962461255119060161, 1.99090073595735865248797190134, 3.12417563612422609833126016120, 5.06926069102626274122332171599, 7.07839011160565786317392193050, 8.707760579324136569571476556836, 9.139766316233918896800355710839, 10.71136176494295565688703647292, 12.21390698074816697036938346230, 12.81386019473007339600902575352, 13.76289707486164334205526210323

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