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Properties

Label 2-960-12.11-c1-0-30
Degree $2$
Conductor $960$
Sign $-0.577 - 0.816i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.41i)3-s + i·5-s − 4.82i·7-s + (−1.00 + 2.82i)9-s − 4.82·11-s − 1.17·13-s + (1.41 − i)15-s + 0.828i·17-s + 6i·19-s + (−6.82 + 4.82i)21-s + 0.828·23-s − 25-s + (5.00 − 1.41i)27-s − 6i·29-s + 2i·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + 0.447i·5-s − 1.82i·7-s + (−0.333 + 0.942i)9-s − 1.45·11-s − 0.324·13-s + (0.365 − 0.258i)15-s + 0.200i·17-s + 1.37i·19-s + (−1.49 + 1.05i)21-s + 0.172·23-s − 0.200·25-s + (0.962 − 0.272i)27-s − 1.11i·29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.577 - 0.816i$
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: \(1\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.41i)T \)
5 \( 1 - iT \)
good7 \( 1 + 4.82iT - 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 0.828T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 6.82T + 37T^{2} \)
41 \( 1 - 9.65iT - 41T^{2} \)
43 \( 1 + 1.17iT - 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 4.48iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.935287955289420757807758393698, −8.138151731897559740460959451231, −7.72677272977581703666703798065, −7.03834173591666465335768003058, −6.18313784298317670521309286599, −5.17470266983475655995497612029, −4.13958936815996914161471974474, −2.88968016490006584835597453052, −1.48010923179670432604690697203, 0, 2.30989687645947812239959880051, 3.23795905039036865832092386929, 4.94128209225941397038737784643, 5.12251178693461889315439714230, 5.96106662217612777163239722611, 7.12003611490561403476145127556, 8.424153321154012899292450692719, 8.957681960230560678512435682091, 9.645812733666428517773047953516

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