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Properties

Label 2-960-24.11-c3-0-47
Degree 22
Conductor 960960
Sign 0.349−0.936i0.349 - 0.936i
Analytic cond. 56.641856.6418
Root an. cond. 7.526077.52607
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(960s/2 ΓC(s) L(s)=((0.349−0.936i) Λ‾(4−s)\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}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((0.349−0.936i) Λ‾(1−s)\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: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.349−0.936i0.349 - 0.936i
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ960(671,⋅)\chi_{960} (671, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :3/2), 0.349−0.936i)(2,\ 960,\ (\ :3/2),\ 0.349 - 0.936i)

Particular Values

L(2)L(2) ≈\approx 2.6503887362.650388736
L(12)L(\frac12) ≈\approx 2.6503887362.650388736
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−2.15−4.72i)T 1 + (-2.15 - 4.72i)T
5 1−5T 1 - 5T
good7 1+8.15iT−343T2 1 + 8.15iT - 343T^{2}
11 1+4.17iT−1.33e3T2 1 + 4.17iT - 1.33e3T^{2}
13 1−26.1iT−2.19e3T2 1 - 26.1iT - 2.19e3T^{2}
17 1+107.iT−4.91e3T2 1 + 107. iT - 4.91e3T^{2}
19 1−89.1T+6.85e3T2 1 - 89.1T + 6.85e3T^{2}
23 1−49.7T+1.21e4T2 1 - 49.7T + 1.21e4T^{2}
29 1−187.T+2.43e4T2 1 - 187.T + 2.43e4T^{2}
31 1−135.iT−2.97e4T2 1 - 135. iT - 2.97e4T^{2}
37 1−407.iT−5.06e4T2 1 - 407. iT - 5.06e4T^{2}
41 1−322.iT−6.89e4T2 1 - 322. iT - 6.89e4T^{2}
43 1−243.T+7.95e4T2 1 - 243.T + 7.95e4T^{2}
47 1+394.T+1.03e5T2 1 + 394.T + 1.03e5T^{2}
53 1−348.T+1.48e5T2 1 - 348.T + 1.48e5T^{2}
59 1+89.5iT−2.05e5T2 1 + 89.5iT - 2.05e5T^{2}
61 1+281.iT−2.26e5T2 1 + 281. iT - 2.26e5T^{2}
67 1+348.T+3.00e5T2 1 + 348.T + 3.00e5T^{2}
71 1+171.T+3.57e5T2 1 + 171.T + 3.57e5T^{2}
73 1−1.03e3T+3.89e5T2 1 - 1.03e3T + 3.89e5T^{2}
79 1−644.iT−4.93e5T2 1 - 644. iT - 4.93e5T^{2}
83 1+727.iT−5.71e5T2 1 + 727. iT - 5.71e5T^{2}
89 1+126.iT−7.04e5T2 1 + 126. iT - 7.04e5T^{2}
97 1−419.T+9.12e5T2 1 - 419.T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line