Properties

Label 2-960-5.4-c1-0-13
Degree $2$
Conductor $960$
Sign $0.894 + 0.447i$
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  + i·3-s + (−1 + 2i)5-s − 4i·7-s − 9-s − 4i·13-s + (−2 − i)15-s + 8·19-s + 4·21-s − 4i·23-s + (−3 − 4i)25-s i·27-s − 6·29-s + 8·31-s + (8 + 4i)35-s − 4i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.447 + 0.894i)5-s − 1.51i·7-s − 0.333·9-s − 1.10i·13-s + (−0.516 − 0.258i)15-s + 1.83·19-s + 0.872·21-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s + 1.43·31-s + (1.35 + 0.676i)35-s − 0.657i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31244 - 0.309826i\)
\(L(\frac12)\) \(\approx\) \(1.31244 - 0.309826i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (1 - 2i)T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−10.15171153648370455755473970473, −9.401050543851123561989668694038, −7.968597947754405242181241195371, −7.58190292405612893024544673975, −6.68985947927623828947056456137, −5.59818820227675093164261046348, −4.46234821772170555779770077073, −3.62190216826083507701274624659, −2.85035100503354657465662311055, −0.71825506655189134218654800495, 1.29914658144467961439861519154, 2.48135447738237884783741304560, 3.75854924307519632440789484748, 5.09376829222111799629919852001, 5.60530398770157714774534466327, 6.70866737316784786177296608315, 7.71240976081536572422566947413, 8.415103588954964488254877030404, 9.262222206835457484755614124135, 9.650792749302337946181799221801

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