Properties

Label 2-76-19.15-c4-0-5
Degree $2$
Conductor $76$
Sign $0.885 + 0.465i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (14.9 − 2.64i)3-s + (1.74 − 1.46i)5-s + (22.6 − 39.2i)7-s + (141. − 51.5i)9-s + (51.9 + 89.9i)11-s + (−220. − 38.9i)13-s + (22.3 − 26.6i)15-s + (−69.3 − 25.2i)17-s + (340. − 118. i)19-s + (235. − 647. i)21-s + (547. + 459. i)23-s + (−107. + 610. i)25-s + (918. − 530. i)27-s + (−295. − 813. i)29-s + (−723. − 417. i)31-s + ⋯
L(s)  = 1  + (1.66 − 0.293i)3-s + (0.0699 − 0.0586i)5-s + (0.462 − 0.800i)7-s + (1.74 − 0.636i)9-s + (0.429 + 0.743i)11-s + (−1.30 − 0.230i)13-s + (0.0992 − 0.118i)15-s + (−0.239 − 0.0873i)17-s + (0.944 − 0.328i)19-s + (0.534 − 1.46i)21-s + (1.03 + 0.867i)23-s + (−0.172 + 0.976i)25-s + (1.26 − 0.727i)27-s + (−0.351 − 0.966i)29-s + (−0.753 − 0.434i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.885 + 0.465i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ 0.885 + 0.465i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.76346 - 0.682165i\)
\(L(\frac12)\) \(\approx\) \(2.76346 - 0.682165i\)
\(L(3)\) 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 \( 1 + (-340. + 118. i)T \)
good3 \( 1 + (-14.9 + 2.64i)T + (76.1 - 27.7i)T^{2} \)
5 \( 1 + (-1.74 + 1.46i)T + (108. - 615. i)T^{2} \)
7 \( 1 + (-22.6 + 39.2i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-51.9 - 89.9i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (220. + 38.9i)T + (2.68e4 + 9.76e3i)T^{2} \)
17 \( 1 + (69.3 + 25.2i)T + (6.39e4 + 5.36e4i)T^{2} \)
23 \( 1 + (-547. - 459. i)T + (4.85e4 + 2.75e5i)T^{2} \)
29 \( 1 + (295. + 813. i)T + (-5.41e5 + 4.54e5i)T^{2} \)
31 \( 1 + (723. + 417. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 603. iT - 1.87e6T^{2} \)
41 \( 1 + (1.04e3 - 183. i)T + (2.65e6 - 9.66e5i)T^{2} \)
43 \( 1 + (2.61e3 - 2.19e3i)T + (5.93e5 - 3.36e6i)T^{2} \)
47 \( 1 + (1.02e3 - 372. i)T + (3.73e6 - 3.13e6i)T^{2} \)
53 \( 1 + (1.04e3 - 1.24e3i)T + (-1.37e6 - 7.77e6i)T^{2} \)
59 \( 1 + (653. - 1.79e3i)T + (-9.28e6 - 7.78e6i)T^{2} \)
61 \( 1 + (-2.68e3 - 2.25e3i)T + (2.40e6 + 1.36e7i)T^{2} \)
67 \( 1 + (1.44e3 + 3.97e3i)T + (-1.54e7 + 1.29e7i)T^{2} \)
71 \( 1 + (3.64e3 + 4.34e3i)T + (-4.41e6 + 2.50e7i)T^{2} \)
73 \( 1 + (337. + 1.91e3i)T + (-2.66e7 + 9.71e6i)T^{2} \)
79 \( 1 + (6.05e3 - 1.06e3i)T + (3.66e7 - 1.33e7i)T^{2} \)
83 \( 1 + (-3.62e3 + 6.28e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (-1.04e4 - 1.84e3i)T + (5.89e7 + 2.14e7i)T^{2} \)
97 \( 1 + (4.97e3 - 1.36e4i)T + (-6.78e7 - 5.69e7i)T^{2} \)
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

−13.69368023507081718669943537037, −13.03114376312127854166001368145, −11.58693096328914897605576725255, −9.873576279742943949785234009873, −9.203944080310662121501112186515, −7.67636510671905947428477720227, −7.23808008307181438359140046561, −4.70867720429921382756042244655, −3.20616952871512378155840686588, −1.64970565001316583867850162744, 2.11993587646237701322159836014, 3.37170180194869652357854658888, 5.04033779840967416421757791930, 7.09738898329044381039028243524, 8.419486984627607097853618769461, 9.052540337743955998403435037951, 10.17728825828138414890647222352, 11.75959199224505750865786244641, 12.97425368122304756792144555131, 14.34310487598645049137554991892

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