↑

Properties

Label 2-960-192.155-c1-0-16
Degree $2$
Conductor $960$
Sign $0.0233 - 0.999i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.834 − 1.14i)2-s + (−0.116 + 1.72i)3-s + (−0.608 − 1.90i)4-s + (−0.980 − 0.195i)5-s + (1.87 + 1.57i)6-s + (−0.720 − 0.298i)7-s + (−2.68 − 0.893i)8-s + (−2.97 − 0.403i)9-s + (−1.04 + 0.957i)10-s + (0.148 − 0.0994i)11-s + (3.36 − 0.829i)12-s + (0.940 + 4.72i)13-s + (−0.941 + 0.573i)14-s + (0.451 − 1.67i)15-s + (−3.25 + 2.31i)16-s + (0.808 + 0.808i)17-s + ⋯
L(s)  = 1  + (0.589 − 0.807i)2-s + (−0.0673 + 0.997i)3-s + (−0.304 − 0.952i)4-s + (−0.438 − 0.0872i)5-s + (0.765 + 0.642i)6-s + (−0.272 − 0.112i)7-s + (−0.948 − 0.316i)8-s + (−0.990 − 0.134i)9-s + (−0.329 + 0.302i)10-s + (0.0448 − 0.0299i)11-s + (0.970 − 0.239i)12-s + (0.260 + 1.31i)13-s + (−0.251 + 0.153i)14-s + (0.116 − 0.431i)15-s + (−0.814 + 0.579i)16-s + (0.196 + 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0233 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0233 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0233 - 0.999i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.0233 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702585 + 0.686398i\)
\(L(\frac12)\) \(\approx\) \(0.702585 + 0.686398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.834 + 1.14i)T \)
3 \( 1 + (0.116 - 1.72i)T \)
5 \( 1 + (0.980 + 0.195i)T \)
good7 \( 1 + (0.720 + 0.298i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.148 + 0.0994i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-0.940 - 4.72i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (-0.808 - 0.808i)T + 17iT^{2} \)
19 \( 1 + (-1.23 - 6.20i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (7.96 - 3.29i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.45 - 2.17i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 - 4.51T + 31T^{2} \)
37 \( 1 + (-1.39 - 0.277i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.198 - 0.478i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.37 - 0.921i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (7.68 - 7.68i)T - 47iT^{2} \)
53 \( 1 + (-0.142 - 0.213i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-0.0352 + 0.177i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (-0.464 + 0.694i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (3.79 + 2.53i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (2.73 - 6.59i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (0.219 + 0.529i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.529 + 0.529i)T - 79iT^{2} \)
83 \( 1 + (3.70 - 0.737i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (-5.62 + 13.5i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + 4.37iT - 97T^{2} \)
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

−10.09884814471570232722960051551, −9.799125005381021143603926264815, −8.862903260977942640451195269727, −7.937522907641823602918227225659, −6.41878872473171481121455510688, −5.74911850823194217507785602244, −4.61866802271152986989366395218, −3.93487947616411312383070647349, −3.25547172667156065681039993440, −1.72324427861568621021895909148, 0.36870319271364033994217952172, 2.54263295392318247710346054059, 3.40855843399910981691233160613, 4.71131697583190990801320735664, 5.70511592294392873031467073163, 6.41617443207669496006126780242, 7.21490550902799958374924032425, 8.031779770490480253733978475205, 8.462804457663875996193066372432, 9.636345109030428413793985104360

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