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Properties

Label 2-960-192.155-c1-0-17
Degree $2$
Conductor $960$
Sign $0.602 - 0.798i$
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  + (−1.22 − 0.699i)2-s + (−1.69 + 0.343i)3-s + (1.02 + 1.71i)4-s + (0.980 + 0.195i)5-s + (2.32 + 0.765i)6-s + (−4.51 − 1.86i)7-s + (−0.0532 − 2.82i)8-s + (2.76 − 1.16i)9-s + (−1.06 − 0.925i)10-s + (0.133 − 0.0895i)11-s + (−2.32 − 2.56i)12-s + (0.874 + 4.39i)13-s + (4.23 + 5.45i)14-s + (−1.73 + 0.00533i)15-s + (−1.91 + 3.51i)16-s + (−4.18 − 4.18i)17-s + ⋯
L(s)  = 1  + (−0.869 − 0.494i)2-s + (−0.980 + 0.198i)3-s + (0.510 + 0.859i)4-s + (0.438 + 0.0872i)5-s + (0.949 + 0.312i)6-s + (−1.70 − 0.706i)7-s + (−0.0188 − 0.999i)8-s + (0.921 − 0.388i)9-s + (−0.338 − 0.292i)10-s + (0.0403 − 0.0269i)11-s + (−0.671 − 0.741i)12-s + (0.242 + 1.21i)13-s + (1.13 + 1.45i)14-s + (−0.447 + 0.00137i)15-s + (−0.478 + 0.878i)16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.602 - 0.798i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376325 + 0.187455i\)
\(L(\frac12)\) \(\approx\) \(0.376325 + 0.187455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.699i)T \)
3 \( 1 + (1.69 - 0.343i)T \)
5 \( 1 + (-0.980 - 0.195i)T \)
good7 \( 1 + (4.51 + 1.86i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.133 + 0.0895i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-0.874 - 4.39i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (4.18 + 4.18i)T + 17iT^{2} \)
19 \( 1 + (0.162 + 0.817i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-7.60 + 3.14i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.34 + 2.01i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 6.08T + 31T^{2} \)
37 \( 1 + (0.153 + 0.0306i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-2.71 - 6.54i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.44 - 4.97i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (3.10 - 3.10i)T - 47iT^{2} \)
53 \( 1 + (-5.56 - 8.33i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (1.38 - 6.96i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (-1.63 + 2.44i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (-9.38 - 6.27i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (1.38 - 3.34i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.02 + 2.47i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-7.75 + 7.75i)T - 79iT^{2} \)
83 \( 1 + (-15.0 + 2.98i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (3.99 - 9.63i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−10.14863387198521739369608243184, −9.274599660278459142508242444059, −9.157037902106448169220809208145, −7.34848421189631805689020178314, −6.63691112353487046981420902249, −6.40592309928977936655706966358, −4.74353113559923266915156388265, −3.75151352125513029634052573290, −2.61780860295572919414014775072, −0.950851974564557886731816197151, 0.37768339627230160921058038282, 1.97985152525221717060567980269, 3.39730410371612576909821271671, 5.27667519955622348073202342261, 5.71439616333996316444243539901, 6.61052924808229014900956321340, 7.02012868093094177468458725535, 8.364148229236904635790531813637, 9.150121330429276201460519219694, 9.887017417903481575750266691430

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