Properties

Label 2-76-19.10-c4-0-6
Degree $2$
Conductor $76$
Sign $-0.602 + 0.798i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.504 − 0.600i)3-s + (−8.75 − 3.18i)5-s + (−3.87 + 6.71i)7-s + (13.9 − 79.1i)9-s + (−94.7 − 164. i)11-s + (−152. + 181. i)13-s + (2.49 + 6.86i)15-s + (−73.3 − 415. i)17-s + (171. − 317. i)19-s + (5.99 − 1.05i)21-s + (−669. + 243. i)23-s + (−412. − 345. i)25-s + (−109. + 63.2i)27-s + (1.55e3 + 274. i)29-s + (−389. − 224. i)31-s + ⋯
L(s)  = 1  + (−0.0560 − 0.0667i)3-s + (−0.350 − 0.127i)5-s + (−0.0791 + 0.137i)7-s + (0.172 − 0.977i)9-s + (−0.782 − 1.35i)11-s + (−0.902 + 1.07i)13-s + (0.0111 + 0.0305i)15-s + (−0.253 − 1.43i)17-s + (0.474 − 0.880i)19-s + (0.0135 − 0.00239i)21-s + (−1.26 + 0.460i)23-s + (−0.659 − 0.553i)25-s + (−0.150 + 0.0868i)27-s + (1.85 + 0.326i)29-s + (−0.404 − 0.233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.798i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.602 + 0.798i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.602 + 0.798i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.375431 - 0.753227i\)
\(L(\frac12)\) \(\approx\) \(0.375431 - 0.753227i\)
\(L(3)\) 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 \( 1 + (-171. + 317. i)T \)
good3 \( 1 + (0.504 + 0.600i)T + (-14.0 + 79.7i)T^{2} \)
5 \( 1 + (8.75 + 3.18i)T + (478. + 401. i)T^{2} \)
7 \( 1 + (3.87 - 6.71i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (94.7 + 164. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (152. - 181. i)T + (-4.95e3 - 2.81e4i)T^{2} \)
17 \( 1 + (73.3 + 415. i)T + (-7.84e4 + 2.85e4i)T^{2} \)
23 \( 1 + (669. - 243. i)T + (2.14e5 - 1.79e5i)T^{2} \)
29 \( 1 + (-1.55e3 - 274. i)T + (6.64e5 + 2.41e5i)T^{2} \)
31 \( 1 + (389. + 224. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 923. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.59e3 - 1.89e3i)T + (-4.90e5 + 2.78e6i)T^{2} \)
43 \( 1 + (-374. - 136. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-189. + 1.07e3i)T + (-4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + (293. + 807. i)T + (-6.04e6 + 5.07e6i)T^{2} \)
59 \( 1 + (2.30e3 - 406. i)T + (1.13e7 - 4.14e6i)T^{2} \)
61 \( 1 + (-5.17e3 + 1.88e3i)T + (1.06e7 - 8.89e6i)T^{2} \)
67 \( 1 + (6.47e3 + 1.14e3i)T + (1.89e7 + 6.89e6i)T^{2} \)
71 \( 1 + (-877. + 2.40e3i)T + (-1.94e7 - 1.63e7i)T^{2} \)
73 \( 1 + (3.63e3 - 3.05e3i)T + (4.93e6 - 2.79e7i)T^{2} \)
79 \( 1 + (7.42e3 + 8.84e3i)T + (-6.76e6 + 3.83e7i)T^{2} \)
83 \( 1 + (-5.15e3 + 8.92e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (-1.40e3 + 1.67e3i)T + (-1.08e7 - 6.17e7i)T^{2} \)
97 \( 1 + (-6.41e3 + 1.13e3i)T + (8.31e7 - 3.02e7i)T^{2} \)
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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

−13.50088033054914136045132748778, −12.03466365437332077954619917457, −11.49639826441566985487649979259, −9.879775675471782000107894775140, −8.867829620348794674777762997698, −7.49254794244850775425008581844, −6.20242355440040226121266861137, −4.62096123716734058713726305842, −2.89954178067924595947451635497, −0.41676589080451704832613323625, 2.24220428016112872330893386005, 4.23848311400182132414791267304, 5.58585417670835756808162562746, 7.42077741514597502380912159984, 8.099523731855887719865415818701, 10.10626531577489934635610709747, 10.47211521318046984313281777916, 12.19469499153552107561873136696, 12.87116319983379403326148022881, 14.21479762846230163391304560376

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