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Properties

Label 2-480-1.1-c3-0-20
Degree 22
Conductor 480480
Sign −1-1
Analytic cond. 28.320928.3209
Root an. cond. 5.321745.32174
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 − 8·7-s + 9·9-s + 4·11-s − 6·13-s − 15·15-s − 2·17-s − 16·19-s − 24·21-s − 60·23-s + 25·25-s + 27·27-s − 142·29-s − 176·31-s + 12·33-s + 40·35-s − 214·37-s − 18·39-s − 278·41-s − 68·43-s − 45·45-s + 116·47-s − 279·49-s − 6·51-s − 350·53-s − 20·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.431·7-s + 1/3·9-s + 0.109·11-s − 0.128·13-s − 0.258·15-s − 0.0285·17-s − 0.193·19-s − 0.249·21-s − 0.543·23-s + 1/5·25-s + 0.192·27-s − 0.909·29-s − 1.01·31-s + 0.0633·33-s + 0.193·35-s − 0.950·37-s − 0.0739·39-s − 1.05·41-s − 0.241·43-s − 0.149·45-s + 0.360·47-s − 0.813·49-s − 0.0164·51-s − 0.907·53-s − 0.0490·55-s + ⋯

Functional equation

Λ(s)=(480s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(480s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 480480    =    25⋅3⋅52^{5} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 28.320928.3209
Root analytic conductor: 5.321745.32174
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 480, ( :3/2), −1)(2,\ 480,\ (\ :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+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
13 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
17 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
19 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
23 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
29 1+142T+p3T2 1 + 142 T + p^{3} T^{2}
31 1+176T+p3T2 1 + 176 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1+278T+p3T2 1 + 278 T + p^{3} T^{2}
43 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
47 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
53 1+350T+p3T2 1 + 350 T + p^{3} T^{2}
59 1−684T+p3T2 1 - 684 T + p^{3} T^{2}
61 1+394T+p3T2 1 + 394 T + p^{3} T^{2}
67 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
71 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
73 1+398T+p3T2 1 + 398 T + p^{3} T^{2}
79 1−136T+p3T2 1 - 136 T + p^{3} T^{2}
83 1−436T+p3T2 1 - 436 T + p^{3} T^{2}
89 1+750T+p3T2 1 + 750 T + p^{3} T^{2}
97 1−82T+p3T2 1 - 82 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

−10.03667066451914679545999361939, −9.200392894119146804153597059233, −8.350513642862924929963843508021, −7.44469801189638006812238885205, −6.57933110198190133414743920865, −5.32538360994380945831121853066, −4.05915449890457009034356986418, −3.20053686185766007100443328749, −1.81841197679123431491000146518, 0, 1.81841197679123431491000146518, 3.20053686185766007100443328749, 4.05915449890457009034356986418, 5.32538360994380945831121853066, 6.57933110198190133414743920865, 7.44469801189638006812238885205, 8.350513642862924929963843508021, 9.200392894119146804153597059233, 10.03667066451914679545999361939

Graph of the ZZ-function along the critical line