↑

Properties

Label 2-960-60.59-c1-0-29
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  + 1.73i·3-s − 2.23·5-s − 2.99·9-s − 3.87i·15-s − 4.47·17-s − 7.74i·19-s + 3.46i·23-s + 5.00·25-s − 5.19i·27-s − 7.74i·31-s + 6.70·45-s − 10.3i·47-s − 7·49-s − 7.74i·51-s − 4.47·53-s + ⋯
L(s)  = 1  + 0.999i·3-s − 0.999·5-s − 0.999·9-s − 1.00i·15-s − 1.08·17-s − 1.77i·19-s + 0.722i·23-s + 1.00·25-s − 0.999i·27-s − 1.39i·31-s + 0.999·45-s − 1.51i·47-s − 49-s − 1.08i·51-s − 0.614·53-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: $\chi_{960} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300001 - 0.300001i\)
\(L(\frac12)\) \(\approx\) \(0.300001 - 0.300001i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 + 2.23T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 7.74iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 4.47T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−9.727939368229263202076913357804, −9.022351723757054852949280641426, −8.328001576792936585237460119625, −7.34363929716500844820843682693, −6.41890865544763685154463608633, −5.15296307328531477240275735327, −4.45852834767557782820453677640, −3.61593322698850644058711388529, −2.54626829222565269812095577583, −0.20034692340401449629368587485, 1.44042876658745361297631698231, 2.81075761345438359158455539923, 3.88797235453881582313118720187, 4.99675659172442238511820460079, 6.21499557041755378041714572793, 6.86126063424023425305756681306, 7.82940575944848630890002843276, 8.304094432170598776640011962829, 9.150023496923248475246736936626, 10.45884441191373988271301589943

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