Properties

Label 2-76-19.5-c1-0-1
Degree $2$
Conductor $76$
Sign $0.731 + 0.681i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.314 − 1.78i)3-s + (0.216 + 0.181i)5-s + (−0.579 − 1.00i)7-s + (−0.268 − 0.0976i)9-s + (−0.622 + 1.07i)11-s + (0.977 + 5.54i)13-s + (0.391 − 0.328i)15-s + (−6.25 + 2.27i)17-s + (3.09 − 3.07i)19-s + (−1.97 + 0.719i)21-s + (−4.65 + 3.90i)23-s + (−0.854 − 4.84i)25-s + (2.46 − 4.26i)27-s + (3.64 + 1.32i)29-s + (−0.0400 − 0.0693i)31-s + ⋯
L(s)  = 1  + (0.181 − 1.03i)3-s + (0.0966 + 0.0811i)5-s + (−0.219 − 0.379i)7-s + (−0.0894 − 0.0325i)9-s + (−0.187 + 0.325i)11-s + (0.270 + 1.53i)13-s + (0.101 − 0.0848i)15-s + (−1.51 + 0.551i)17-s + (0.709 − 0.704i)19-s + (−0.431 + 0.156i)21-s + (−0.969 + 0.813i)23-s + (−0.170 − 0.969i)25-s + (0.473 − 0.820i)27-s + (0.677 + 0.246i)29-s + (−0.00719 − 0.0124i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910901 - 0.358361i\)
\(L(\frac12)\) \(\approx\) \(0.910901 - 0.358361i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.09 + 3.07i)T \)
good3 \( 1 + (-0.314 + 1.78i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-0.216 - 0.181i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.579 + 1.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.622 - 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.977 - 5.54i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (6.25 - 2.27i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.65 - 3.90i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.64 - 1.32i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.0400 + 0.0693i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.71T + 37T^{2} \)
41 \( 1 + (-1.11 + 6.33i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.189 - 0.158i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (10.8 + 3.96i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-3.50 + 2.93i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (9.32 - 3.39i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.27 - 3.58i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.47 + 1.26i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.14 - 5.99i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.191 + 1.08i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.50 - 8.55i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-5.77 - 9.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.418 + 2.37i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (13.4 - 4.90i)T + (74.3 - 62.3i)T^{2} \)
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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

−13.90769136641589653350141559525, −13.52147255782843814186809253933, −12.33586797664166016324066429251, −11.28338047658485229925074222421, −9.863309749290262918442455837486, −8.550121792616364526495426663169, −7.19471116148253859542615680615, −6.43228785279641891613148961709, −4.32098435597146296935726102364, −2.03174298614485429115003541067, 3.14077216174695863572963621086, 4.70999931251830686835701656751, 6.08904163186243579542048942217, 7.951002098614999342042326151235, 9.182533985978052170063299917715, 10.13157097724070052780820995248, 11.11689836821432454535423985245, 12.55771129393488143890874920686, 13.59837305535996901654013569814, 14.92964552969141908199504401619

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