Properties

Label 2-960-192.11-c1-0-50
Degree $2$
Conductor $960$
Sign $-0.943 - 0.331i$
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  + (1.25 + 0.642i)2-s + (0.399 + 1.68i)3-s + (1.17 + 1.61i)4-s + (−0.831 + 0.555i)5-s + (−0.578 + 2.38i)6-s + (−0.496 + 1.19i)7-s + (0.439 + 2.79i)8-s + (−2.68 + 1.34i)9-s + (−1.40 + 0.165i)10-s + (−0.701 + 3.52i)11-s + (−2.25 + 2.62i)12-s + (2.14 − 3.21i)13-s + (−1.39 + 1.19i)14-s + (−1.26 − 1.17i)15-s + (−1.24 + 3.80i)16-s + (−0.789 − 0.789i)17-s + ⋯
L(s)  = 1  + (0.890 + 0.454i)2-s + (0.230 + 0.972i)3-s + (0.587 + 0.809i)4-s + (−0.371 + 0.248i)5-s + (−0.236 + 0.971i)6-s + (−0.187 + 0.452i)7-s + (0.155 + 0.987i)8-s + (−0.893 + 0.449i)9-s + (−0.444 + 0.0524i)10-s + (−0.211 + 1.06i)11-s + (−0.651 + 0.758i)12-s + (0.595 − 0.891i)13-s + (−0.372 + 0.318i)14-s + (−0.327 − 0.304i)15-s + (−0.310 + 0.950i)16-s + (−0.191 − 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413745 + 2.42245i\)
\(L(\frac12)\) \(\approx\) \(0.413745 + 2.42245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.642i)T \)
3 \( 1 + (-0.399 - 1.68i)T \)
5 \( 1 + (0.831 - 0.555i)T \)
good7 \( 1 + (0.496 - 1.19i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.701 - 3.52i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.14 + 3.21i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 + (0.789 + 0.789i)T + 17iT^{2} \)
19 \( 1 + (-4.26 + 6.38i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-1.68 - 4.07i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (7.54 - 1.50i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + 2.10T + 31T^{2} \)
37 \( 1 + (-3.92 + 2.62i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.29 - 0.534i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.68 - 8.46i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + (-8.06 + 8.06i)T - 47iT^{2} \)
53 \( 1 + (0.329 + 0.0655i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (-6.16 - 9.23i)T + (-22.5 + 54.5i)T^{2} \)
61 \( 1 + (-2.71 + 0.539i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.02 - 15.1i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (-0.620 - 0.256i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.564 - 0.233i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-10.9 + 10.9i)T - 79iT^{2} \)
83 \( 1 + (12.8 + 8.55i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (-2.34 - 0.973i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 3.06iT - 97T^{2} \)
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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

−10.52793842489572017876556336294, −9.456350272075416677452321120340, −8.772036916471417721907882166122, −7.65619477544470220511836900652, −7.11212310791190050391741010933, −5.73753815758526167679661449878, −5.19221140232705300471037162952, −4.22291986903900126564673709950, −3.29789374296777408441080089516, −2.49878021018113475955082640811, 0.836261033344786489994571781849, 2.01648970069499176995722106011, 3.37696574947163719789307605958, 3.94925105587749988398494351549, 5.41794939871618046539920465848, 6.12961937772274738086765014561, 6.97400115754910682932903889531, 7.84243110971682778255287245968, 8.747288833626608891778940136054, 9.711976617324401518008617874024

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