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Properties

Label 2-960-20.7-c1-0-17
Degree $2$
Conductor $960$
Sign $-0.197 + 0.980i$
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  + (−0.707 − 0.707i)3-s + (−2.23 + 0.0743i)5-s + (3.16 − 3.16i)7-s + 1.00i·9-s + 4.46i·11-s + (2.51 − 2.51i)13-s + (1.63 + 1.52i)15-s + (−2.30 − 2.30i)17-s + 2.61·19-s − 4.46·21-s + (−1.64 − 1.64i)23-s + (4.98 − 0.332i)25-s + (0.707 − 0.707i)27-s − 8.17i·29-s + 4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.999 + 0.0332i)5-s + (1.19 − 1.19i)7-s + 0.333i·9-s + 1.34i·11-s + (0.698 − 0.698i)13-s + (0.421 + 0.394i)15-s + (−0.560 − 0.560i)17-s + 0.600·19-s − 0.975·21-s + (−0.342 − 0.342i)23-s + (0.997 − 0.0664i)25-s + (0.136 − 0.136i)27-s − 1.51i·29-s + 0.718i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703746 - 0.859477i\)
\(L(\frac12)\) \(\approx\) \(0.703746 - 0.859477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.23 - 0.0743i)T \)
good7 \( 1 + (-3.16 + 3.16i)T - 7iT^{2} \)
11 \( 1 - 4.46iT - 11T^{2} \)
13 \( 1 + (-2.51 + 2.51i)T - 13iT^{2} \)
17 \( 1 + (2.30 + 2.30i)T + 17iT^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + (1.64 + 1.64i)T + 23iT^{2} \)
29 \( 1 + 8.17iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (5.80 + 5.80i)T + 37iT^{2} \)
41 \( 1 + 2.61T + 41T^{2} \)
43 \( 1 + (5.14 + 5.14i)T + 43iT^{2} \)
47 \( 1 + (0.679 - 0.679i)T - 47iT^{2} \)
53 \( 1 + (-7.81 + 7.81i)T - 53iT^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-9.44 + 9.44i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (5.61 - 5.61i)T - 73iT^{2} \)
79 \( 1 - 3.57T + 79T^{2} \)
83 \( 1 + (-1.34 - 1.34i)T + 83iT^{2} \)
89 \( 1 + 17.6iT - 89T^{2} \)
97 \( 1 + (1.32 + 1.32i)T + 97iT^{2} \)
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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

−10.11554480867543963573246022045, −8.706497817599969886070746937079, −7.85125885345038956262024146903, −7.38631043797840178535730385539, −6.67910141701796140906758451523, −5.18482845237374683992105777920, −4.52026210868024488349177390245, −3.64879607809458234361577461822, −1.93546302058491607081430974221, −0.59854138231452478268799427676, 1.44073051550183293578680039389, 3.09460265800754781096187210456, 4.07223776046739447925817503558, 5.06014328972212870665761501087, 5.78614924605767960986165757100, 6.79202809380029193145178164342, 8.063731809879568541342581687618, 8.575549994594952268886533131640, 9.126776187764919180585604854745, 10.57821360115196087652278996867

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