↑

Properties

Label 2-76-19.18-c10-0-8
Degree $2$
Conductor $76$
Sign $-0.442 - 0.896i$
Analytic cond. $48.2871$
Root an. cond. $6.94889$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 367. i·3-s + 5.51e3·5-s − 5.76e3·7-s − 7.59e4·9-s + 1.76e5·11-s + 1.53e5i·13-s + 2.02e6i·15-s − 5.71e5·17-s + (1.09e6 + 2.22e6i)19-s − 2.11e6i·21-s + 9.05e6·23-s + 2.06e7·25-s − 6.19e6i·27-s − 2.02e7i·29-s + 1.67e7i·31-s + ⋯
L(s)  = 1  + 1.51i·3-s + 1.76·5-s − 0.342·7-s − 1.28·9-s + 1.09·11-s + 0.414i·13-s + 2.66i·15-s − 0.402·17-s + (0.442 + 0.896i)19-s − 0.518i·21-s + 1.40·23-s + 2.11·25-s − 0.431i·27-s − 0.986i·29-s + 0.584i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(48.2871\)
Root analytic conductor: \(6.94889\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5),\ -0.442 - 0.896i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.59504 + 2.56650i\)
\(L(\frac12)\) \(\approx\) \(1.59504 + 2.56650i\)
\(L(6)\) 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 \( 1 + (-1.09e6 - 2.22e6i)T \)
good3 \( 1 - 367. iT - 5.90e4T^{2} \)
5 \( 1 - 5.51e3T + 9.76e6T^{2} \)
7 \( 1 + 5.76e3T + 2.82e8T^{2} \)
11 \( 1 - 1.76e5T + 2.59e10T^{2} \)
13 \( 1 - 1.53e5iT - 1.37e11T^{2} \)
17 \( 1 + 5.71e5T + 2.01e12T^{2} \)
23 \( 1 - 9.05e6T + 4.14e13T^{2} \)
29 \( 1 + 2.02e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.67e7iT - 8.19e14T^{2} \)
37 \( 1 + 3.35e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.23e8iT - 1.34e16T^{2} \)
43 \( 1 - 3.71e6T + 2.16e16T^{2} \)
47 \( 1 + 6.38e7T + 5.25e16T^{2} \)
53 \( 1 - 6.53e8iT - 1.74e17T^{2} \)
59 \( 1 + 8.11e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.81e8T + 7.13e17T^{2} \)
67 \( 1 - 4.81e7iT - 1.82e18T^{2} \)
71 \( 1 - 3.60e9iT - 3.25e18T^{2} \)
73 \( 1 + 4.28e8T + 4.29e18T^{2} \)
79 \( 1 + 4.91e9iT - 9.46e18T^{2} \)
83 \( 1 + 4.84e9T + 1.55e19T^{2} \)
89 \( 1 + 8.02e9iT - 3.11e19T^{2} \)
97 \( 1 + 8.61e9iT - 7.37e19T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.95943191365555013747147982659, −11.36118957642416059779748863549, −10.22724579717749841823322642055, −9.535710659685074386905984802970, −8.943586719990069340922030519559, −6.59407182686734864253137549565, −5.58905095472455222929737158587, −4.42020847255939442592755746532, −3.03641499045375847800282211856, −1.47611905020330650083313704549, 0.855470392775650179467989999284, 1.73199069085474776358014532454, 2.85121495731654534665334190905, 5.30546940370337580053671527431, 6.46501084344087020372441589482, 7.00351662112345456294538766737, 8.757000998320608894502507455638, 9.635864942309820132565939581181, 11.09370058813284267474739625907, 12.44905426747210372083738880341

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