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Properties

Label 2-76-19.12-c2-0-0
Degree $2$
Conductor $76$
Sign $-0.995 + 0.0955i$
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  + (−3.30 + 1.90i)3-s + (−1.55 − 2.69i)5-s − 11.1·7-s + (2.78 − 4.82i)9-s − 5.69·11-s + (−6.74 − 3.89i)13-s + (10.3 + 5.94i)15-s + (9.15 + 15.8i)17-s + (17.9 + 6.15i)19-s + (36.9 − 21.3i)21-s + (−17.0 + 29.5i)23-s + (7.64 − 13.2i)25-s − 13.0i·27-s + (−33.1 − 19.1i)29-s − 10.4i·31-s + ⋯
L(s)  = 1  + (−1.10 + 0.636i)3-s + (−0.311 − 0.539i)5-s − 1.59·7-s + (0.309 − 0.536i)9-s − 0.517·11-s + (−0.518 − 0.299i)13-s + (0.686 + 0.396i)15-s + (0.538 + 0.932i)17-s + (0.946 + 0.324i)19-s + (1.76 − 1.01i)21-s + (−0.742 + 1.28i)23-s + (0.305 − 0.529i)25-s − 0.484i·27-s + (−1.14 − 0.660i)29-s − 0.337i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.995 + 0.0955i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.995 + 0.0955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00399846 - 0.0834959i\)
\(L(\frac12)\) \(\approx\) \(0.00399846 - 0.0834959i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-17.9 - 6.15i)T \)
good3 \( 1 + (3.30 - 1.90i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.55 + 2.69i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 11.1T + 49T^{2} \)
11 \( 1 + 5.69T + 121T^{2} \)
13 \( 1 + (6.74 + 3.89i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-9.15 - 15.8i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (17.0 - 29.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (33.1 + 19.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 10.4iT - 961T^{2} \)
37 \( 1 - 30.1iT - 1.36e3T^{2} \)
41 \( 1 + (33.0 - 19.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (29.7 + 51.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (30.9 - 53.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (45.3 + 26.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-73.7 + 42.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.82 + 3.16i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (96.7 + 55.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (55.4 - 32.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-37.7 - 65.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-85.4 + 49.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 60.0T + 6.88e3T^{2} \)
89 \( 1 + (-17.4 - 10.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (50.1 - 28.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−15.24201677334820479267129449687, −13.45635781621800615736876105483, −12.49362108184400826681917284887, −11.63321544991083909239825348700, −10.20328933423221619426634074157, −9.667616625800916601884343150486, −7.86980772491789751987466625109, −6.20671015914353978075435341845, −5.22522306731669629025335248215, −3.59532810758179594771195317076, 0.07684826419554992421680565784, 3.14408625605025496730782139212, 5.37743402605894775116847953514, 6.63931450156731838267623149296, 7.32736217762256298257971434255, 9.384594782019000994195762830685, 10.49215868188841317354111810507, 11.69366722395970038052873278819, 12.48264853129476265023400248848, 13.40914898692451653214785560905

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