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Properties

Label 2-76-4.3-c2-0-12
Degree $2$
Conductor $76$
Sign $-0.600 + 0.799i$
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  + (−1.89 − 0.632i)2-s − 5.34i·3-s + (3.19 + 2.40i)4-s + 5.79·5-s + (−3.38 + 10.1i)6-s − 5.87i·7-s + (−4.55 − 6.57i)8-s − 19.5·9-s + (−10.9 − 3.66i)10-s + 8.07i·11-s + (12.8 − 17.0i)12-s − 14.0·13-s + (−3.71 + 11.1i)14-s − 30.9i·15-s + (4.47 + 15.3i)16-s + 29.9·17-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)2-s − 1.78i·3-s + (0.799 + 0.600i)4-s + 1.15·5-s + (−0.563 + 1.68i)6-s − 0.839i·7-s + (−0.568 − 0.822i)8-s − 2.17·9-s + (−1.09 − 0.366i)10-s + 0.734i·11-s + (1.06 − 1.42i)12-s − 1.07·13-s + (−0.265 + 0.796i)14-s − 2.06i·15-s + (0.279 + 0.960i)16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.600 + 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.415582 - 0.831453i\)
\(L(\frac12)\) \(\approx\) \(0.415582 - 0.831453i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.89 + 0.632i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 5.34iT - 9T^{2} \)
5 \( 1 - 5.79T + 25T^{2} \)
7 \( 1 + 5.87iT - 49T^{2} \)
11 \( 1 - 8.07iT - 121T^{2} \)
13 \( 1 + 14.0T + 169T^{2} \)
17 \( 1 - 29.9T + 289T^{2} \)
23 \( 1 - 8.74iT - 529T^{2} \)
29 \( 1 - 13.4T + 841T^{2} \)
31 \( 1 + 7.65iT - 961T^{2} \)
37 \( 1 - 25.1T + 1.36e3T^{2} \)
41 \( 1 - 49.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2iT - 1.84e3T^{2} \)
47 \( 1 - 46.8iT - 2.20e3T^{2} \)
53 \( 1 + 14.1T + 2.80e3T^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 31.1T + 3.72e3T^{2} \)
67 \( 1 - 34.5iT - 4.48e3T^{2} \)
71 \( 1 + 66.0iT - 5.04e3T^{2} \)
73 \( 1 + 27.5T + 5.32e3T^{2} \)
79 \( 1 - 40.9iT - 6.24e3T^{2} \)
83 \( 1 + 87.8iT - 6.88e3T^{2} \)
89 \( 1 - 28.6T + 7.92e3T^{2} \)
97 \( 1 - 9.35T + 9.40e3T^{2} \)
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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

−13.63126644965022193006085812365, −12.63637970385176286066739818848, −11.92562288293722383063026453770, −10.36932702755899078334385459764, −9.413672861997431809683677520023, −7.73473474469285692476307801533, −7.21300566302059653428119896037, −5.93278891316526139316557196965, −2.50844610645954649694131252476, −1.18649868569568905180848082206, 2.80497305752348175112421955489, 5.24539027922627983342565951823, 5.96196846265594007795345986338, 8.218407574065842333686051952400, 9.444679317546106714968812254587, 9.808413450140216365014253577691, 10.78743482820009645488925370248, 12.06027484347590400747157002855, 14.29326181686783753350890677420, 14.74378085162896993465477666588

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