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Properties

Label 2-960-24.11-c1-0-26
Degree $2$
Conductor $960$
Sign $0.138 + 0.990i$
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  + (1.59 − 0.675i)3-s − 5-s − 0.648i·7-s + (2.08 − 2.15i)9-s − 4.17i·11-s + 0.847i·13-s + (−1.59 + 0.675i)15-s − 2.91i·17-s − 0.847·19-s + (−0.438 − 1.03i)21-s + 2.34·23-s + 25-s + (1.86 − 4.84i)27-s − 6.87·29-s − 2.17i·31-s + ⋯
L(s)  = 1  + (0.920 − 0.390i)3-s − 0.447·5-s − 0.244i·7-s + (0.695 − 0.718i)9-s − 1.25i·11-s + 0.235i·13-s + (−0.411 + 0.174i)15-s − 0.706i·17-s − 0.194·19-s + (−0.0955 − 0.225i)21-s + 0.488·23-s + 0.200·25-s + (0.359 − 0.933i)27-s − 1.27·29-s − 0.390i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.138 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43543 - 1.24843i\)
\(L(\frac12)\) \(\approx\) \(1.43543 - 1.24843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.59 + 0.675i)T \)
5 \( 1 + T \)
good7 \( 1 + 0.648iT - 7T^{2} \)
11 \( 1 + 4.17iT - 11T^{2} \)
13 \( 1 - 0.847iT - 13T^{2} \)
17 \( 1 + 2.91iT - 17T^{2} \)
19 \( 1 + 0.847T + 19T^{2} \)
23 \( 1 - 2.34T + 23T^{2} \)
29 \( 1 + 6.87T + 29T^{2} \)
31 \( 1 + 2.17iT - 31T^{2} \)
37 \( 1 + 5.53iT - 37T^{2} \)
41 \( 1 - 4.31iT - 41T^{2} \)
43 \( 1 - 9.56T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 6.87iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + 5.43T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 - 10.6iT - 83T^{2} \)
89 \( 1 - 17.9iT - 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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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

−9.489001270771407236899920788759, −9.041933788826042985002150574059, −8.087267838613018987668882356148, −7.48819192103294186838299292928, −6.62025220478954212580653362587, −5.56086184393748020459862787666, −4.20300920816886018100469494624, −3.42834580364161982501770848781, −2.39792248403203499706000056379, −0.821487594602944756684450909407, 1.76712075535428858844657747461, 2.87202658392886088985065304752, 3.98575765517602218023962980596, 4.67090031494839303319038926600, 5.84402306364231474227811457963, 7.24611325831607212998045988236, 7.60286885578645609811533525562, 8.734055572980106138072835939911, 9.201846102027817440456449490377, 10.23046950953794054885814586221

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