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Properties

Label 2-960-5.4-c3-0-68
Degree $2$
Conductor $960$
Sign $-0.999 - 0.0160i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (0.178 − 11.1i)5-s − 35.0i·7-s − 9·9-s − 25.6·11-s + 37.6i·13-s + (33.5 + 0.536i)15-s − 95.7i·17-s + 50.8·19-s + 105.·21-s − 110. i·23-s + (−124. − 4i)25-s − 27i·27-s − 54.5·29-s + 198.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.0160 − 0.999i)5-s − 1.89i·7-s − 0.333·9-s − 0.702·11-s + 0.803i·13-s + (0.577 + 0.00923i)15-s − 1.36i·17-s + 0.614·19-s + 1.09·21-s − 1.00i·23-s + (−0.999 − 0.0320i)25-s − 0.192i·27-s − 0.349·29-s + 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0160i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.999 - 0.0160i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8970970307\)
\(L(\frac12)\) \(\approx\) \(0.8970970307\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 + (-0.178 + 11.1i)T \)
good7 \( 1 + 35.0iT - 343T^{2} \)
11 \( 1 + 25.6T + 1.33e3T^{2} \)
13 \( 1 - 37.6iT - 2.19e3T^{2} \)
17 \( 1 + 95.7iT - 4.91e3T^{2} \)
19 \( 1 - 50.8T + 6.85e3T^{2} \)
23 \( 1 + 110. iT - 1.21e4T^{2} \)
29 \( 1 + 54.5T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 + 266. iT - 5.06e4T^{2} \)
41 \( 1 - 103.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 - 597. iT - 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 + 223.T + 2.05e5T^{2} \)
61 \( 1 + 485.T + 2.26e5T^{2} \)
67 \( 1 - 876. iT - 3.00e5T^{2} \)
71 \( 1 - 585.T + 3.57e5T^{2} \)
73 \( 1 - 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 + 685.T + 4.93e5T^{2} \)
83 \( 1 - 305. iT - 5.71e5T^{2} \)
89 \( 1 + 887.T + 7.04e5T^{2} \)
97 \( 1 + 556. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.503107133563289856300618843674, −8.377817074078845434961107539890, −7.57506007086961423574320321824, −6.82173474593088831284943526072, −5.51080071241078138719220741387, −4.49852494971327762959357801024, −4.22197117415714484262611280425, −2.83315352689586680227219239077, −1.15710929027036725883454087288, −0.23820130380906611051906225258, 1.74797225225809062209640747076, 2.67808413607816863972215345814, 3.37492841695897209515294619006, 5.18663533659423481190396093977, 5.86423273286072830583365177404, 6.48523255203081336021150763756, 7.73784231777088848057390462844, 8.186515457468282068586006874569, 9.175080359184590366453340289597, 10.11266375889886116470986421652

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