Properties

Label 2-960-192.35-c1-0-126
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: $\chi_{960} (611, \cdot )$
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

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

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