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Properties

Label 2-960-12.11-c1-0-17
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

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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

\(L(1)\) \(\approx\) \(0.607382 - 0.607382i\)
\(L(\frac12)\) \(\approx\) \(0.607382 - 0.607382i\)
\(L(\frac{3}{2})\) 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 \( 1 - 10T + 97T^{2} \)
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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

−10.21575383366636883243627158122, −9.210618041996372748661058847873, −7.84226868960334334557665886840, −7.11497971684968336468316248350, −6.50209597120957223609352888581, −5.59029648855747847446993202119, −4.27254604993901083543535084823, −3.89865217245323185442279801352, −2.00009676375686446543975626198, −0.48937519566131699730116159891, 1.35594937719992004067689183599, 2.74519630418242092018695059058, 4.30505036211368661504363219478, 5.11488404002943330331590815223, 5.81992161310353618861476197595, 6.69837545135997129217770093922, 7.60346132926526585045388629238, 8.737159929881840478544578006087, 9.527545412813170081752027252551, 10.03507598530524887247208713824

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