↑

Properties

Label 2-960-12.11-c1-0-27
Degree $2$
Conductor $960$
Sign $i$
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  + 1.73·3-s − i·5-s − 3.46i·7-s + 2.99·9-s − 3.46·11-s − 4·13-s − 1.73i·15-s − 6i·17-s − 3.46i·19-s − 5.99i·21-s + 3.46·23-s − 25-s + 5.19·27-s + 6i·29-s − 3.46i·31-s + ⋯
L(s)  = 1  + 1.00·3-s − 0.447i·5-s − 1.30i·7-s + 0.999·9-s − 1.04·11-s − 1.10·13-s − 0.447i·15-s − 1.45i·17-s − 0.794i·19-s − 1.30i·21-s + 0.722·23-s − 0.200·25-s + 1.00·27-s + 1.11i·29-s − 0.622i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34665 - 1.34665i\)
\(L(\frac12)\) \(\approx\) \(1.34665 - 1.34665i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 + iT \)
good7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 6.92iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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

−9.691867954171618712453493261694, −9.123025386945200333661827517791, −7.974445668820064879739830568487, −7.43062228153555787117059164152, −6.82580653174917820999052156720, −5.00500413134931241950736903349, −4.62727704328125151185581494693, −3.29956783434892709884503642743, −2.41587507820056681957513959122, −0.76714289168996611660210428909, 2.06121599581559811849482216811, 2.66395661926230098667515205010, 3.75193665852114912527313561294, 5.01015137904076881958405818697, 5.91290086632325466704977949110, 6.99383880967085274439688156584, 7.992100747755734457317828607006, 8.394267356969108770247602610660, 9.433976523693604584959277105583, 10.06319064993950468425522078020

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