Properties

Label 2-76-76.23-c4-0-16
Degree $2$
Conductor $76$
Sign $-0.0701 + 0.997i$
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  + (−3.15 − 2.45i)2-s + (−8.88 + 1.56i)3-s + (3.93 + 15.5i)4-s + (−25.4 + 21.3i)5-s + (31.9 + 16.8i)6-s + (18.4 + 10.6i)7-s + (25.6 − 58.6i)8-s + (0.390 − 0.142i)9-s + (132. − 4.92i)10-s + (14.9 − 8.65i)11-s + (−59.3 − 131. i)12-s + (4.76 − 26.9i)13-s + (−32.0 − 78.8i)14-s + (192. − 229. i)15-s + (−224. + 122. i)16-s + (−392. − 142. i)17-s + ⋯
L(s)  = 1  + (−0.789 − 0.613i)2-s + (−0.987 + 0.174i)3-s + (0.246 + 0.969i)4-s + (−1.01 + 0.853i)5-s + (0.886 + 0.468i)6-s + (0.376 + 0.217i)7-s + (0.400 − 0.916i)8-s + (0.00481 − 0.00175i)9-s + (1.32 − 0.0492i)10-s + (0.123 − 0.0715i)11-s + (−0.411 − 0.914i)12-s + (0.0281 − 0.159i)13-s + (−0.163 − 0.402i)14-s + (0.856 − 1.02i)15-s + (−0.878 + 0.477i)16-s + (−1.35 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0701 + 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0701 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0701 + 0.997i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.0701 + 0.997i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.236441 - 0.253644i\)
\(L(\frac12)\) \(\approx\) \(0.236441 - 0.253644i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.15 + 2.45i)T \)
19 \( 1 + (-299. + 201. i)T \)
good3 \( 1 + (8.88 - 1.56i)T + (76.1 - 27.7i)T^{2} \)
5 \( 1 + (25.4 - 21.3i)T + (108. - 615. i)T^{2} \)
7 \( 1 + (-18.4 - 10.6i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-14.9 + 8.65i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-4.76 + 26.9i)T + (-2.68e4 - 9.76e3i)T^{2} \)
17 \( 1 + (392. + 142. i)T + (6.39e4 + 5.36e4i)T^{2} \)
23 \( 1 + (-390. + 465. i)T + (-4.85e4 - 2.75e5i)T^{2} \)
29 \( 1 + (-20.3 + 7.39i)T + (5.41e5 - 4.54e5i)T^{2} \)
31 \( 1 + (457. + 264. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.09e3T + 1.87e6T^{2} \)
41 \( 1 + (-152. - 866. i)T + (-2.65e6 + 9.66e5i)T^{2} \)
43 \( 1 + (-1.53e3 - 1.82e3i)T + (-5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (990. + 2.72e3i)T + (-3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 + (1.42e3 + 1.19e3i)T + (1.37e6 + 7.77e6i)T^{2} \)
59 \( 1 + (1.21e3 - 3.33e3i)T + (-9.28e6 - 7.78e6i)T^{2} \)
61 \( 1 + (-2.08e3 - 1.74e3i)T + (2.40e6 + 1.36e7i)T^{2} \)
67 \( 1 + (802. + 2.20e3i)T + (-1.54e7 + 1.29e7i)T^{2} \)
71 \( 1 + (3.77e3 + 4.50e3i)T + (-4.41e6 + 2.50e7i)T^{2} \)
73 \( 1 + (1.36e3 + 7.75e3i)T + (-2.66e7 + 9.71e6i)T^{2} \)
79 \( 1 + (555. - 97.9i)T + (3.66e7 - 1.33e7i)T^{2} \)
83 \( 1 + (7.25e3 + 4.19e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-1.34e3 + 7.60e3i)T + (-5.89e7 - 2.14e7i)T^{2} \)
97 \( 1 + (-3.75e3 - 1.36e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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

−13.10891752784393689804061363517, −11.59795578946714968603828169850, −11.42413908974695841960702375741, −10.54322315721367242606936975485, −9.006642444967538431410459950810, −7.69780484786944715264302409741, −6.57765150588099247889075878365, −4.57498251157277496832137181123, −2.85044961291370607594959188731, −0.32614539085184886450889706581, 1.06958007028158189153846172425, 4.52764878456403678058316538966, 5.75076999806672438472521347439, 7.12533018149315425278013853191, 8.229347630041931072738477457987, 9.299002953500883344665036409554, 10.99387884022913743898834376202, 11.51125026596249987056154922445, 12.72773972589054965989260891800, 14.30102813347454056654549678993

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