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Properties

Label 2-960-192.83-c1-0-6
Degree $2$
Conductor $960$
Sign $-0.845 + 0.533i$
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  + (−1.27 + 0.614i)2-s + (0.216 + 1.71i)3-s + (1.24 − 1.56i)4-s + (−0.980 + 0.195i)5-s + (−1.33 − 2.05i)6-s + (2.39 − 0.992i)7-s + (−0.624 + 2.75i)8-s + (−2.90 + 0.743i)9-s + (1.12 − 0.850i)10-s + (−1.60 − 1.06i)11-s + (2.95 + 1.80i)12-s + (0.531 − 2.67i)13-s + (−2.44 + 2.73i)14-s + (−0.547 − 1.64i)15-s + (−0.898 − 3.89i)16-s + (−2.20 + 2.20i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.434i)2-s + (0.124 + 0.992i)3-s + (0.622 − 0.782i)4-s + (−0.438 + 0.0872i)5-s + (−0.543 − 0.839i)6-s + (0.905 − 0.375i)7-s + (−0.220 + 0.975i)8-s + (−0.968 + 0.247i)9-s + (0.357 − 0.269i)10-s + (−0.482 − 0.322i)11-s + (0.854 + 0.520i)12-s + (0.147 − 0.741i)13-s + (−0.652 + 0.731i)14-s + (−0.141 − 0.424i)15-s + (−0.224 − 0.974i)16-s + (−0.534 + 0.534i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.845 + 0.533i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0829670 - 0.286792i\)
\(L(\frac12)\) \(\approx\) \(0.0829670 - 0.286792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.614i)T \)
3 \( 1 + (-0.216 - 1.71i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
good7 \( 1 + (-2.39 + 0.992i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.60 + 1.06i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (-0.531 + 2.67i)T + (-12.0 - 4.97i)T^{2} \)
17 \( 1 + (2.20 - 2.20i)T - 17iT^{2} \)
19 \( 1 + (1.40 - 7.06i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (6.47 + 2.68i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.54 - 3.80i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + 9.85T + 31T^{2} \)
37 \( 1 + (4.73 - 0.942i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (4.81 - 11.6i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.10 - 2.74i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 + (1.51 + 1.51i)T + 47iT^{2} \)
53 \( 1 + (-4.07 + 6.09i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-0.0928 - 0.466i)T + (-54.5 + 22.5i)T^{2} \)
61 \( 1 + (4.35 + 6.51i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (7.43 - 4.96i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (-2.52 - 6.10i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.24 - 3.01i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.90 - 2.90i)T + 79iT^{2} \)
83 \( 1 + (-5.30 - 1.05i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (5.90 + 14.2i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−10.43145130864931765929887730026, −9.839223953362425135525003938444, −8.455816767742172203511657743555, −8.351472527999356538124850835335, −7.52781264875570337892773344367, −6.18883406228503501607929664830, −5.41849686957867921805452801492, −4.41365841130443207456755065972, −3.31007100623336129966114546999, −1.80581004414724780774001076732, 0.17210521635478041242463896462, 1.81791924976270830307897642086, 2.46038819904811309980179285000, 3.90671329111339771503837016031, 5.19125848002204927017263809591, 6.48846985216941196480700876649, 7.35326733342972793047344311273, 7.79453341000390880609838046854, 8.939241605327155883661444270372, 9.014848661935631715282710905542

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