Properties

Degree $2$
Conductor $960$
Sign $0.447 - 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + (−2 − i)5-s + 2i·7-s − 9-s + 2·11-s + 2i·13-s + (−1 + 2i)15-s + 6i·17-s − 8·19-s + 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·29-s − 2i·33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (−0.258 + 0.516i)15-s + 1.45i·17-s − 1.83·19-s + 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.48·29-s − 0.348i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.447 - 0.894i$
Motivic weight: \(1\)
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820510 + 0.507103i\)
\(L(\frac12)\) \(\approx\) \(0.820510 + 0.507103i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2 + i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−10.26870671570928651544769329884, −8.938326430419058969539559281165, −8.524708963394712764702786462144, −7.88366258694318355537920330272, −6.51494980513937177174276862019, −6.25788656631927173474389913661, −4.73018368441145937045797029115, −4.04529255878406932176264212297, −2.66281176407388149517619154979, −1.38493726723817759877125434325, 0.47867887922774400161696325136, 2.58222617141465845178748427393, 3.75625188599162295958960767058, 4.29048960202738611119589370171, 5.40303945019765498410426269798, 6.69776241199813181284086333673, 7.25810532710205171025243442797, 8.263121450869900726581895515883, 9.005182915485349800402921961099, 10.08363037580864116350997436415

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