Properties

Label 2-76-19.14-c6-0-8
Degree $2$
Conductor $76$
Sign $0.0331 + 0.999i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.4 + 2.89i)3-s + (43.3 + 36.3i)5-s + (−212. − 367. i)7-s + (−423. − 154. i)9-s + (793. − 1.37e3i)11-s + (−340. + 60.0i)13-s + (607. + 723. i)15-s + (−904. + 329. i)17-s + (4.84e3 − 4.85e3i)19-s + (−2.42e3 − 6.65e3i)21-s + (325. − 273. i)23-s + (−2.15e3 − 1.22e4i)25-s + (−1.70e4 − 9.84e3i)27-s + (−1.75e3 + 4.80e3i)29-s + (2.41e4 − 1.39e4i)31-s + ⋯
L(s)  = 1  + (0.608 + 0.107i)3-s + (0.346 + 0.291i)5-s + (−0.618 − 1.07i)7-s + (−0.580 − 0.211i)9-s + (0.596 − 1.03i)11-s + (−0.155 + 0.0273i)13-s + (0.179 + 0.214i)15-s + (−0.184 + 0.0670i)17-s + (0.707 − 0.707i)19-s + (−0.261 − 0.718i)21-s + (0.0267 − 0.0224i)23-s + (−0.138 − 0.782i)25-s + (−0.866 − 0.500i)27-s + (−0.0717 + 0.197i)29-s + (0.809 − 0.467i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0331 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0331 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.0331 + 0.999i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ 0.0331 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.28587 - 1.24389i\)
\(L(\frac12)\) \(\approx\) \(1.28587 - 1.24389i\)
\(L(4)\) 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 \( 1 + (-4.84e3 + 4.85e3i)T \)
good3 \( 1 + (-16.4 - 2.89i)T + (685. + 249. i)T^{2} \)
5 \( 1 + (-43.3 - 36.3i)T + (2.71e3 + 1.53e4i)T^{2} \)
7 \( 1 + (212. + 367. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-793. + 1.37e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (340. - 60.0i)T + (4.53e6 - 1.65e6i)T^{2} \)
17 \( 1 + (904. - 329. i)T + (1.84e7 - 1.55e7i)T^{2} \)
23 \( 1 + (-325. + 273. i)T + (2.57e7 - 1.45e8i)T^{2} \)
29 \( 1 + (1.75e3 - 4.80e3i)T + (-4.55e8 - 3.82e8i)T^{2} \)
31 \( 1 + (-2.41e4 + 1.39e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + 5.22e4iT - 2.56e9T^{2} \)
41 \( 1 + (-4.41e4 - 7.77e3i)T + (4.46e9 + 1.62e9i)T^{2} \)
43 \( 1 + (9.68e4 + 8.12e4i)T + (1.09e9 + 6.22e9i)T^{2} \)
47 \( 1 + (2.12e4 + 7.74e3i)T + (8.25e9 + 6.92e9i)T^{2} \)
53 \( 1 + (-9.01e4 - 1.07e5i)T + (-3.84e9 + 2.18e10i)T^{2} \)
59 \( 1 + (-9.90e4 - 2.72e5i)T + (-3.23e10 + 2.71e10i)T^{2} \)
61 \( 1 + (2.14e5 - 1.79e5i)T + (8.94e9 - 5.07e10i)T^{2} \)
67 \( 1 + (1.79e5 - 4.92e5i)T + (-6.92e10 - 5.81e10i)T^{2} \)
71 \( 1 + (-1.06e5 + 1.26e5i)T + (-2.22e10 - 1.26e11i)T^{2} \)
73 \( 1 + (-2.49e4 + 1.41e5i)T + (-1.42e11 - 5.17e10i)T^{2} \)
79 \( 1 + (-5.68e5 - 1.00e5i)T + (2.28e11 + 8.31e10i)T^{2} \)
83 \( 1 + (-2.88e5 - 4.99e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (5.72e5 - 1.00e5i)T + (4.67e11 - 1.69e11i)T^{2} \)
97 \( 1 + (6.91e3 + 1.89e4i)T + (-6.38e11 + 5.35e11i)T^{2} \)
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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

−13.54686574212397728420958262053, −11.85622583636777162836058104398, −10.69887919616018686384025196804, −9.584973672117230895039186408976, −8.556814615955113743283961225411, −7.11562657955220248878554214838, −5.94615107146769510084919167881, −3.93996838371173107631612989012, −2.79841007487702205739228682133, −0.61825788046692074501191893824, 1.85436159428400257912119750688, 3.17848741785041785861389233425, 5.11141343525995206182299162142, 6.40829212783641098453129104270, 7.959925150320965399244090215490, 9.140666058571504605390642332992, 9.803578567702860482249910092746, 11.61212726658076076507604428883, 12.50909667474423548990460494212, 13.56993459396158228816575751737

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