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Properties

Label 2-76-76.7-c2-0-6
Degree $2$
Conductor $76$
Sign $0.968 + 0.249i$
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  + (−1.69 − 1.05i)2-s + (3.11 − 1.79i)3-s + (1.77 + 3.58i)4-s + (4.52 + 7.84i)5-s + (−7.19 − 0.233i)6-s + 2.81i·7-s + (0.776 − 7.96i)8-s + (1.97 − 3.41i)9-s + (0.586 − 18.1i)10-s − 11.6i·11-s + (11.9 + 7.98i)12-s + (−2.20 + 3.81i)13-s + (2.97 − 4.78i)14-s + (28.2 + 16.2i)15-s + (−9.72 + 12.7i)16-s + (−9.59 − 16.6i)17-s + ⋯
L(s)  = 1  + (−0.849 − 0.527i)2-s + (1.03 − 0.599i)3-s + (0.442 + 0.896i)4-s + (0.905 + 1.56i)5-s + (−1.19 − 0.0388i)6-s + 0.402i·7-s + (0.0970 − 0.995i)8-s + (0.218 − 0.379i)9-s + (0.0586 − 1.81i)10-s − 1.06i·11-s + (0.997 + 0.665i)12-s + (−0.169 + 0.293i)13-s + (0.212 − 0.342i)14-s + (1.88 + 1.08i)15-s + (−0.607 + 0.794i)16-s + (−0.564 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.968 + 0.249i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.968 + 0.249i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26290 - 0.159786i\)
\(L(\frac12)\) \(\approx\) \(1.26290 - 0.159786i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.69 + 1.05i)T \)
19 \( 1 + (-9.06 + 16.6i)T \)
good3 \( 1 + (-3.11 + 1.79i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.52 - 7.84i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.81iT - 49T^{2} \)
11 \( 1 + 11.6iT - 121T^{2} \)
13 \( 1 + (2.20 - 3.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (9.59 + 16.6i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (25.2 + 14.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.9 + 27.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 23.9iT - 961T^{2} \)
37 \( 1 - 19.3T + 1.36e3T^{2} \)
41 \( 1 + (-2.87 - 4.97i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (12.4 - 7.18i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.4 + 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-16.2 + 28.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-9.13 + 5.27i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.5 - 92.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (109. + 63.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (3.89 - 2.24i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (12.5 + 21.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-70.4 + 40.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 138. iT - 6.88e3T^{2} \)
89 \( 1 + (24.1 - 41.8i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (26.7 + 46.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.86956234573437556297097831069, −13.52269795458776320530052281917, −11.76301004695044563864471468355, −10.78112398193332042711312185942, −9.636226319824880911389847412745, −8.635512303795456973733011411235, −7.38493804543163554420168758229, −6.37720513436191684042646298612, −3.02417820175235250413614031457, −2.31584498916007716170249702199, 1.77373411870289718757230455109, 4.46644273049010459124764665714, 5.90314208026575918239566179226, 7.81740589194215845795320647503, 8.733521814844136474771858974508, 9.627451945157087111720628913402, 10.19169972789596160747443272368, 12.26643332032251165934204641265, 13.51791275909984631496590960901, 14.47846394105124241292735258232

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