↑

Properties

Label 2-960-120.83-c0-0-1
Degree $2$
Conductor $960$
Sign $0.973 - 0.229i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $0$
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 + (0.707 − 0.707i)5-s + (1 + i)7-s − 1.00i·9-s − 1.41i·11-s + 1.00i·15-s − 1.41·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·29-s + (1.00 + 1.00i)33-s + 1.41·35-s + (−0.707 − 0.707i)45-s + i·49-s + (1.41 + 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (1 + i)7-s − 1.00i·9-s − 1.41i·11-s + 1.00i·15-s − 1.41·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·29-s + (1.00 + 1.00i)33-s + 1.41·35-s + (−0.707 − 0.707i)45-s + i·49-s + (1.41 + 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002112375\)
\(L(\frac12)\) \(\approx\) \(1.002112375\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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

−10.43786079021992187716107514302, −9.157112863590228275312620846363, −8.921453427937425923878671134124, −8.033475421901706582529018648620, −6.50990431398882370894712363142, −5.59355330656827680041490061431, −5.30968278210664876462823136780, −4.28016959484032349516212535533, −2.90831671921819073599019488408, −1.36362610022627662750103896327, 1.49758369882584317256068604853, 2.38586201262363086365378115423, 4.17225653656596031189787227386, 5.01475695429214899445319288834, 5.99573898227353712830389586530, 6.98851401177520020255977309786, 7.37690000027550811358983236535, 8.246638583808154282489067482193, 9.718356404207933394410395676345, 10.26965421527073542207716056227

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