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Properties

Label 2-960-24.11-c3-0-47
Degree $2$
Conductor $960$
Sign $0.349 - 0.936i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 + 4.72i)3-s + 5·5-s − 8.15i·7-s + (−17.6 + 20.3i)9-s − 4.17i·11-s + 26.1i·13-s + (10.7 + 23.6i)15-s − 107. i·17-s + 89.1·19-s + (38.5 − 17.5i)21-s + 49.7·23-s + 25·25-s + (−134. − 39.6i)27-s + 187.·29-s + 135. i·31-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + 0.447·5-s − 0.440i·7-s + (−0.655 + 0.755i)9-s − 0.114i·11-s + 0.557i·13-s + (0.185 + 0.406i)15-s − 1.53i·17-s + 1.07·19-s + (0.400 − 0.182i)21-s + 0.451·23-s + 0.200·25-s + (−0.959 − 0.282i)27-s + 1.20·29-s + 0.783i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.349 - 0.936i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.650388736\)
\(L(\frac12)\) \(\approx\) \(2.650388736\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.15 - 4.72i)T \)
5 \( 1 - 5T \)
good7 \( 1 + 8.15iT - 343T^{2} \)
11 \( 1 + 4.17iT - 1.33e3T^{2} \)
13 \( 1 - 26.1iT - 2.19e3T^{2} \)
17 \( 1 + 107. iT - 4.91e3T^{2} \)
19 \( 1 - 89.1T + 6.85e3T^{2} \)
23 \( 1 - 49.7T + 1.21e4T^{2} \)
29 \( 1 - 187.T + 2.43e4T^{2} \)
31 \( 1 - 135. iT - 2.97e4T^{2} \)
37 \( 1 - 407. iT - 5.06e4T^{2} \)
41 \( 1 - 322. iT - 6.89e4T^{2} \)
43 \( 1 - 243.T + 7.95e4T^{2} \)
47 \( 1 + 394.T + 1.03e5T^{2} \)
53 \( 1 - 348.T + 1.48e5T^{2} \)
59 \( 1 + 89.5iT - 2.05e5T^{2} \)
61 \( 1 + 281. iT - 2.26e5T^{2} \)
67 \( 1 + 348.T + 3.00e5T^{2} \)
71 \( 1 + 171.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 - 644. iT - 4.93e5T^{2} \)
83 \( 1 + 727. iT - 5.71e5T^{2} \)
89 \( 1 + 126. iT - 7.04e5T^{2} \)
97 \( 1 - 419.T + 9.12e5T^{2} \)
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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

−9.756239260158594241511524430939, −9.176195120502105358901541363967, −8.291846575360765169259008736627, −7.31258718791412637452113979412, −6.40859015317191665812509305458, −5.11011888922106702859209918599, −4.68385857369919516948857045956, −3.37179695661819461095279182071, −2.63063704893593190189158316941, −1.05363502155787139106088585975, 0.75013853924014134254938154622, 1.90566164916035556797731808706, 2.82030245837907961703791212606, 3.91939125501531757974404151038, 5.46473013588349903675262610910, 5.99872112024832940596192390004, 7.00880287464541336449692306785, 7.81499545203969280545845694234, 8.616118526160948332697621414157, 9.295713348879400393458129904136

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