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Properties

Label 2-960-1.1-c3-0-37
Degree 22
Conductor 960960
Sign −1-1
Analytic cond. 56.641856.6418
Root an. cond. 7.526077.52607
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5·5-s − 16·7-s + 9·9-s + 28·11-s + 26·13-s − 15·15-s − 62·17-s + 68·19-s − 48·21-s − 208·23-s + 25·25-s + 27·27-s + 58·29-s + 160·31-s + 84·33-s + 80·35-s − 270·37-s + 78·39-s + 282·41-s − 76·43-s − 45·45-s − 280·47-s − 87·49-s − 186·51-s + 210·53-s − 140·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.863·7-s + 1/3·9-s + 0.767·11-s + 0.554·13-s − 0.258·15-s − 0.884·17-s + 0.821·19-s − 0.498·21-s − 1.88·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.926·31-s + 0.443·33-s + 0.386·35-s − 1.19·37-s + 0.320·39-s + 1.07·41-s − 0.269·43-s − 0.149·45-s − 0.868·47-s − 0.253·49-s − 0.510·51-s + 0.544·53-s − 0.343·55-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 960, ( :3/2), −1)(2,\ 960,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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−pT 1 - p T
5 1+pT 1 + p T
good7 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1−28T+p3T2 1 - 28 T + p^{3} T^{2}
13 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
17 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
19 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
23 1+208T+p3T2 1 + 208 T + p^{3} T^{2}
29 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
31 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
37 1+270T+p3T2 1 + 270 T + p^{3} T^{2}
41 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
43 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
47 1+280T+p3T2 1 + 280 T + p^{3} T^{2}
53 1−210T+p3T2 1 - 210 T + p^{3} T^{2}
59 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
61 1+742T+p3T2 1 + 742 T + p^{3} T^{2}
67 1+836T+p3T2 1 + 836 T + p^{3} T^{2}
71 1+504T+p3T2 1 + 504 T + p^{3} T^{2}
73 1+1062T+p3T2 1 + 1062 T + p^{3} T^{2}
79 1−768T+p3T2 1 - 768 T + p^{3} T^{2}
83 1−1052T+p3T2 1 - 1052 T + p^{3} T^{2}
89 1+726T+p3T2 1 + 726 T + p^{3} T^{2}
97 1+1406T+p3T2 1 + 1406 T + p^{3} T^{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

−9.198995920129466640909176038576, −8.458099965942092326622574584209, −7.61113537146148828273944719726, −6.64606337476278320033133388873, −6.00346406290803355038183812630, −4.51317083632006878510715406625, −3.73950047516576550533127127069, −2.86977552578676001773957574844, −1.50865854654502688425095991645, 0, 1.50865854654502688425095991645, 2.86977552578676001773957574844, 3.73950047516576550533127127069, 4.51317083632006878510715406625, 6.00346406290803355038183812630, 6.64606337476278320033133388873, 7.61113537146148828273944719726, 8.458099965942092326622574584209, 9.198995920129466640909176038576

Graph of the ZZ-function along the critical line