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Properties

Label 2-960-120.83-c0-0-0
Degree $2$
Conductor $960$
Sign $0.973 - 0.229i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + (1 + i)7-s − 1.00i·9-s + 1.41i·11-s + 1.00i·15-s + 1.41·21-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·29-s + (1.00 + 1.00i)33-s − 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + (−1.41 − 1.41i)53-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + (1 + i)7-s − 1.00i·9-s + 1.41i·11-s + 1.00i·15-s + 1.41·21-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·29-s + (1.00 + 1.00i)33-s − 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + (−1.41 − 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227994826\)
\(L(\frac12)\) \(\approx\) \(1.227994826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.14698668171008857243783444928, −9.325327957327753353416309111029, −8.332421104760213192260179701343, −7.82975004096773231860833967859, −7.07570279825073309684510405466, −6.19956968992625556355415097497, −4.91320859610938787640206580759, −3.90538046480815627138016934296, −2.62285585087757386117156023543, −1.88817274450980251163900290468, 1.32312159903917246102550372418, 3.12005835630262607581159194448, 3.97586843559610315052751947091, 4.71683592653831103775026280805, 5.57002898946988153715983844440, 7.14991789855500357755990759875, 7.949404544341803078370053048967, 8.498814603955777229115623225043, 9.139610230471446349653445292439, 10.29452669782892380090668272234

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