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Properties

Label 2-76-76.75-c5-0-46
Degree $2$
Conductor $76$
Sign $-0.704 + 0.710i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.06 − 2.51i)2-s + 9.05·3-s + (19.3 − 25.5i)4-s − 89.6·5-s + (45.8 − 22.8i)6-s − 100. i·7-s + (33.4 − 177. i)8-s − 161.·9-s + (−453. + 225. i)10-s − 417. i·11-s + (174. − 231. i)12-s − 221. i·13-s + (−254. − 510. i)14-s − 811.·15-s + (−278. − 985. i)16-s + 902.·17-s + ⋯
L(s)  = 1  + (0.895 − 0.445i)2-s + 0.580·3-s + (0.603 − 0.797i)4-s − 1.60·5-s + (0.519 − 0.258i)6-s − 0.777i·7-s + (0.184 − 0.982i)8-s − 0.662·9-s + (−1.43 + 0.713i)10-s − 1.04i·11-s + (0.350 − 0.463i)12-s − 0.363i·13-s + (−0.346 − 0.696i)14-s − 0.930·15-s + (−0.272 − 0.962i)16-s + 0.757·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.704 + 0.710i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.704 + 0.710i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.855749 - 2.05369i\)
\(L(\frac12)\) \(\approx\) \(0.855749 - 2.05369i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.06 + 2.51i)T \)
19 \( 1 + (-1.55e3 - 209. i)T \)
good3 \( 1 - 9.05T + 243T^{2} \)
5 \( 1 + 89.6T + 3.12e3T^{2} \)
7 \( 1 + 100. iT - 1.68e4T^{2} \)
11 \( 1 + 417. iT - 1.61e5T^{2} \)
13 \( 1 + 221. iT - 3.71e5T^{2} \)
17 \( 1 - 902.T + 1.41e6T^{2} \)
23 \( 1 - 3.96e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.18e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.61e3T + 2.86e7T^{2} \)
37 \( 1 - 1.55e4iT - 6.93e7T^{2} \)
41 \( 1 + 3.21e3iT - 1.15e8T^{2} \)
43 \( 1 + 5.05e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.68e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.74e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.03e4T + 7.14e8T^{2} \)
61 \( 1 - 4.72e4T + 8.44e8T^{2} \)
67 \( 1 - 3.83e4T + 1.35e9T^{2} \)
71 \( 1 + 4.09e4T + 1.80e9T^{2} \)
73 \( 1 - 6.03e3T + 2.07e9T^{2} \)
79 \( 1 + 2.52e4T + 3.07e9T^{2} \)
83 \( 1 + 2.71e3iT - 3.93e9T^{2} \)
89 \( 1 + 1.06e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.16e5iT - 8.58e9T^{2} \)
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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

−13.34805560223066609975735516854, −11.67905005471448516895009280859, −11.49389205301796498709073127524, −9.999195903540113029341426899204, −8.244225299670795378971182409800, −7.35522095329565636492461308381, −5.53805616694163912821720952721, −3.81980116388463199360604018088, −3.21007201435692696367455672533, −0.67773965355042888980209184692, 2.70262278150761273724988128620, 3.91269944258173371881312209982, 5.25655132092262644714725662600, 7.04277107579513114026645566270, 7.971394952475061863337623096461, 8.962609379561330390833559578219, 11.09951514065079014656795480498, 12.09341629200417490878134874541, 12.63325151787103185236823321745, 14.49877590233981625764843395618

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