↑

Properties

Label 2-76-76.11-c2-0-17
Degree $2$
Conductor $76$
Sign $-0.994 - 0.101i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.789 − 1.83i)2-s + (−3.65 − 2.11i)3-s + (−2.75 − 2.90i)4-s + (−1.06 + 1.84i)5-s + (−6.76 + 5.05i)6-s + 1.82i·7-s + (−7.50 + 2.77i)8-s + (4.42 + 7.65i)9-s + (2.55 + 3.42i)10-s − 13.8i·11-s + (3.94 + 16.4i)12-s + (−9.95 − 17.2i)13-s + (3.35 + 1.43i)14-s + (7.80 − 4.50i)15-s + (−0.826 + 15.9i)16-s + (3.83 − 6.63i)17-s + ⋯
L(s)  = 1  + (0.394 − 0.918i)2-s + (−1.21 − 0.703i)3-s + (−0.688 − 0.725i)4-s + (−0.213 + 0.369i)5-s + (−1.12 + 0.842i)6-s + 0.260i·7-s + (−0.938 + 0.346i)8-s + (0.491 + 0.850i)9-s + (0.255 + 0.342i)10-s − 1.25i·11-s + (0.329 + 1.36i)12-s + (−0.765 − 1.32i)13-s + (0.239 + 0.102i)14-s + (0.520 − 0.300i)15-s + (−0.0516 + 0.998i)16-s + (0.225 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.994 - 0.101i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.994 - 0.101i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0354277 + 0.693211i\)
\(L(\frac12)\) \(\approx\) \(0.0354277 + 0.693211i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.789 + 1.83i)T \)
19 \( 1 + (-16.7 + 9.04i)T \)
good3 \( 1 + (3.65 + 2.11i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.06 - 1.84i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 1.82iT - 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 + (9.95 + 17.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-3.83 + 6.63i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (9.14 - 5.28i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.0147 + 0.0255i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 42.2iT - 961T^{2} \)
37 \( 1 - 19.5T + 1.36e3T^{2} \)
41 \( 1 + (-15.7 + 27.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-51.3 - 29.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (77.5 - 44.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (29.8 + 51.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (63.9 + 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.6 + 40.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.77 - 3.33i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-24.1 - 13.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (46.2 - 80.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-110. - 63.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 88.9iT - 6.88e3T^{2} \)
89 \( 1 + (-31.2 - 54.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-64.8 + 112. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−13.29129349193259507635720231460, −12.47055037320261131076155899815, −11.44560668889554689408068939362, −10.95376663894320842989992755462, −9.549203271603088020153348395288, −7.73110331607055842060066230865, −6.06264225932595380499310549915, −5.22887431042082597593092903205, −3.05559685728940352279397694561, −0.61652854025098143434612788386, 4.28271658090919043225218375051, 4.97646083540642332458945232243, 6.38720109819317263962293395968, 7.55553039062788413703664465749, 9.254893442040718110577892424534, 10.29715474131631292041456309199, 11.95182031590234516434793789851, 12.35836178788841053853773446117, 14.00330277398577114135265219561, 14.97647035420843670441857189193

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