Properties

Label 2-76-19.18-c4-0-0
Degree $2$
Conductor $76$
Sign $-0.885 + 0.464i$
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  + 10.5i·3-s − 16.8·5-s − 38.6·7-s − 31.1·9-s − 70.8·11-s − 231. i·13-s − 178. i·15-s − 90.4·17-s + (−319. + 167. i)19-s − 409. i·21-s − 285.·23-s − 342.·25-s + 527. i·27-s + 285. i·29-s + 852. i·31-s + ⋯
L(s)  = 1  + 1.17i·3-s − 0.672·5-s − 0.788·7-s − 0.384·9-s − 0.585·11-s − 1.36i·13-s − 0.791i·15-s − 0.312·17-s + (−0.885 + 0.464i)19-s − 0.928i·21-s − 0.539·23-s − 0.547·25-s + 0.723i·27-s + 0.340i·29-s + 0.886i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.885 + 0.464i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0633840 - 0.257492i\)
\(L(\frac12)\) \(\approx\) \(0.0633840 - 0.257492i\)
\(L(3)\) 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 \( 1 + (319. - 167. i)T \)
good3 \( 1 - 10.5iT - 81T^{2} \)
5 \( 1 + 16.8T + 625T^{2} \)
7 \( 1 + 38.6T + 2.40e3T^{2} \)
11 \( 1 + 70.8T + 1.46e4T^{2} \)
13 \( 1 + 231. iT - 2.85e4T^{2} \)
17 \( 1 + 90.4T + 8.35e4T^{2} \)
23 \( 1 + 285.T + 2.79e5T^{2} \)
29 \( 1 - 285. iT - 7.07e5T^{2} \)
31 \( 1 - 852. iT - 9.23e5T^{2} \)
37 \( 1 + 54.8iT - 1.87e6T^{2} \)
41 \( 1 + 37.4iT - 2.82e6T^{2} \)
43 \( 1 - 691.T + 3.41e6T^{2} \)
47 \( 1 - 2.63e3T + 4.87e6T^{2} \)
53 \( 1 - 3.83e3iT - 7.89e6T^{2} \)
59 \( 1 - 3.33e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.87e3T + 1.38e7T^{2} \)
67 \( 1 - 4.05e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.31e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.54e3T + 2.83e7T^{2} \)
79 \( 1 + 1.03e4iT - 3.89e7T^{2} \)
83 \( 1 + 3.54e3T + 4.74e7T^{2} \)
89 \( 1 + 1.32e4iT - 6.27e7T^{2} \)
97 \( 1 + 4.28e3iT - 8.85e7T^{2} \)
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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

−14.74950813513829660176427555464, −13.20032761822566803960060175081, −12.24190169438572911176211412970, −10.68672282609788604624290366783, −10.18001929595896627997168867891, −8.852858990326135310209763223777, −7.57388684701246057325189250157, −5.81122963488803458951286820947, −4.34940345660256481872129593530, −3.17090468470234517188846609694, 0.12864074262186600805070008438, 2.19370753229739653064038748069, 4.15543513150220231475125543116, 6.25529918198481176347630640529, 7.15824741785784744691476254349, 8.254643720862422231952712118985, 9.652674119096762551192884048963, 11.22030079684402263864707303921, 12.20531326986528623615469092223, 13.07222267280321525673363674377

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