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Properties

Label 2-960-192.155-c1-0-111
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: $\chi_{960} (731, \cdot )$
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

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

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