↑

Properties

Label 2-960-15.14-c2-0-68
Degree $2$
Conductor $960$
Sign $0.262 + 0.964i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
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  + (2.23 − 2i)3-s + (2.23 − 4.47i)5-s + 8i·7-s + (1.00 − 8.94i)9-s − 8.94i·11-s + 12i·13-s + (−3.94 − 14.4i)15-s + 31.3·17-s + 6·19-s + (16 + 17.8i)21-s − 4.47·23-s + (−15.0 − 20.0i)25-s + (−15.6 − 22.0i)27-s − 26.8i·29-s + 34·31-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)3-s + (0.447 − 0.894i)5-s + 1.14i·7-s + (0.111 − 0.993i)9-s − 0.813i·11-s + 0.923i·13-s + (−0.262 − 0.964i)15-s + 1.84·17-s + 0.315·19-s + (0.761 + 0.851i)21-s − 0.194·23-s + (−0.600 − 0.800i)25-s + (−0.579 − 0.814i)27-s − 0.925i·29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.262 + 0.964i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.262 + 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.832076277\)
\(L(\frac12)\) \(\approx\) \(2.832076277\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.23 + 2i)T \)
5 \( 1 + (-2.23 + 4.47i)T \)
good7 \( 1 - 8iT - 49T^{2} \)
11 \( 1 + 8.94iT - 121T^{2} \)
13 \( 1 - 12iT - 169T^{2} \)
17 \( 1 - 31.3T + 289T^{2} \)
19 \( 1 - 6T + 361T^{2} \)
23 \( 1 + 4.47T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 - 34T + 961T^{2} \)
37 \( 1 + 44iT - 1.36e3T^{2} \)
41 \( 1 + 17.8iT - 1.68e3T^{2} \)
43 \( 1 + 28iT - 1.84e3T^{2} \)
47 \( 1 + 4.47T + 2.20e3T^{2} \)
53 \( 1 - 40.2T + 2.80e3T^{2} \)
59 \( 1 - 98.3iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 92iT - 4.48e3T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 56iT - 5.32e3T^{2} \)
79 \( 1 + 78T + 6.24e3T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 + 32iT - 9.40e3T^{2} \)
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

−9.383703482780418009655398150578, −8.790886787133422050141053102679, −8.197943605218226322416271503093, −7.28242800407654945793282028742, −5.93673826602685692966156389750, −5.67508958949851939013615455102, −4.22346908756825975731201470879, −3.01771645480501717207903725595, −2.01777939475810063091984434468, −0.912202730598136773421408566477, 1.39202085132706537602814630350, 2.90660100690840256913061010752, 3.46555676299445214722497939247, 4.60795925441670949770758038089, 5.56524428843464940915362749845, 6.80374586130060162338882849518, 7.62113241363397114043822428921, 8.120538363970398832666854193792, 9.537862662159324290508575197417, 10.14527224795131423992656204403

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