Properties

Label 2-76-19.6-c3-0-0
Degree $2$
Conductor $76$
Sign $-0.633 - 0.773i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.29 − 1.20i)3-s + (−0.0631 − 0.357i)5-s + (−13.5 + 23.4i)7-s + (−11.2 − 9.43i)9-s + (28.4 + 49.2i)11-s + (−72.8 + 26.5i)13-s + (−0.221 + 1.25i)15-s + (−57.2 + 48.0i)17-s + (28.0 − 77.9i)19-s + (72.9 − 61.2i)21-s + (−1.58 + 8.99i)23-s + (117. − 42.7i)25-s + (73.1 + 126. i)27-s + (−165. − 139. i)29-s + (31.6 − 54.8i)31-s + ⋯
L(s)  = 1  + (−0.634 − 0.231i)3-s + (−0.00564 − 0.0320i)5-s + (−0.732 + 1.26i)7-s + (−0.416 − 0.349i)9-s + (0.779 + 1.35i)11-s + (−1.55 + 0.565i)13-s + (−0.00381 + 0.0216i)15-s + (−0.816 + 0.685i)17-s + (0.339 − 0.940i)19-s + (0.758 − 0.636i)21-s + (−0.0143 + 0.0815i)23-s + (0.938 − 0.341i)25-s + (0.521 + 0.903i)27-s + (−1.06 − 0.891i)29-s + (0.183 − 0.317i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -0.633 - 0.773i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.228975 + 0.483364i\)
\(L(\frac12)\) \(\approx\) \(0.228975 + 0.483364i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-28.0 + 77.9i)T \)
good3 \( 1 + (3.29 + 1.20i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (0.0631 + 0.357i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (13.5 - 23.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-28.4 - 49.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (72.8 - 26.5i)T + (1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (57.2 - 48.0i)T + (853. - 4.83e3i)T^{2} \)
23 \( 1 + (1.58 - 8.99i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (165. + 139. i)T + (4.23e3 + 2.40e4i)T^{2} \)
31 \( 1 + (-31.6 + 54.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 109.T + 5.06e4T^{2} \)
41 \( 1 + (-55.2 - 20.0i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 + (-58.1 - 329. i)T + (-7.47e4 + 2.71e4i)T^{2} \)
47 \( 1 + (266. + 223. i)T + (1.80e4 + 1.02e5i)T^{2} \)
53 \( 1 + (-30.0 + 170. i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (647. - 543. i)T + (3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (114. - 647. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-95.7 - 80.3i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-65.1 - 369. i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 + (-81.9 - 29.8i)T + (2.98e5 + 2.50e5i)T^{2} \)
79 \( 1 + (-512. - 186. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (464. - 804. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-1.23e3 + 448. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (-1.10e3 + 925. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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

−14.75164733437219527497557162180, −12.92038847278132593298088171164, −12.18431653585339990846298283595, −11.48022525783916388990991515789, −9.689981727160731278171720369460, −9.027411746599025820695746011718, −7.08034874784945633130818228295, −6.13958963192407702145559041042, −4.68690156680169105493519896485, −2.43147978171155106403672103698, 0.34949978909090756841855170391, 3.32652369386048007431548183598, 4.99326983582783032791385827487, 6.39763810450052713296261319260, 7.61418183174060606226741836078, 9.253812525759328145480072755713, 10.46642167085601284735054934635, 11.22703721788929996712299481734, 12.48780023410992219333068580903, 13.71999902197045615808674136417

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