↑

Properties

Label 2-960-20.3-c1-0-4
Degree $2$
Conductor $960$
Sign $-0.0372 - 0.999i$
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  + (0.707 − 0.707i)3-s + (−0.432 + 2.19i)5-s + (0.611 + 0.611i)7-s − 1.00i·9-s + 5.12i·11-s + (−1.76 − 1.76i)13-s + (1.24 + 1.85i)15-s + (−3.76 + 3.76i)17-s + 1.22·19-s + 0.864·21-s + (−1.07 + 1.07i)23-s + (−4.62 − 1.89i)25-s + (−0.707 − 0.707i)27-s − 0.864i·29-s + 7.81i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.193 + 0.981i)5-s + (0.231 + 0.231i)7-s − 0.333i·9-s + 1.54i·11-s + (−0.488 − 0.488i)13-s + (0.321 + 0.479i)15-s + (−0.912 + 0.912i)17-s + 0.280·19-s + 0.188·21-s + (−0.224 + 0.224i)23-s + (−0.925 − 0.379i)25-s + (−0.136 − 0.136i)27-s − 0.160i·29-s + 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0372 - 0.999i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.0372 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975501 + 1.01254i\)
\(L(\frac12)\) \(\approx\) \(0.975501 + 1.01254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.432 - 2.19i)T \)
good7 \( 1 + (-0.611 - 0.611i)T + 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 17iT^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (1.07 - 1.07i)T - 23iT^{2} \)
29 \( 1 + 0.864iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (6.20 - 6.20i)T - 43iT^{2} \)
47 \( 1 + (-2.29 - 2.29i)T + 47iT^{2} \)
53 \( 1 + (-2.62 - 2.62i)T + 53iT^{2} \)
59 \( 1 - 0.528T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 + (-6.20 - 6.20i)T + 67iT^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 + (2.25 + 2.25i)T + 73iT^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + (-7.95 + 7.95i)T - 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (-0.793 + 0.793i)T - 97iT^{2} \)
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

−10.20785969964482400094253930002, −9.501850225039034262588583963698, −8.433609516305412147115485078457, −7.60232310292840247683185202293, −7.00398222976847185226328675245, −6.18668217057431893675905845286, −4.91835465192631220875631130607, −3.87997434244252744040757970252, −2.69804626784334573469339363528, −1.84488375531372252459802222226, 0.60574249991803581517969219062, 2.27731786024340751865339763629, 3.57268529982197304628751651550, 4.48236486702042843389093539720, 5.26190651992418382398238574838, 6.30527467705901465699715198337, 7.54376968298032051593250086226, 8.254247551503721338327144863816, 9.060862483145166240538990271184, 9.507913556777255613261978038335

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