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Properties

Label 2-23520-1.1-c1-0-47
Degree 22
Conductor 2352023520
Sign −1-1
Analytic cond. 187.808187.808
Root an. cond. 13.704313.7043
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 2·11-s + 2·13-s − 15-s + 4·19-s + 25-s + 27-s − 4·29-s + 6·31-s + 2·33-s − 8·37-s + 2·39-s − 10·41-s − 10·43-s − 45-s − 6·47-s + 10·53-s − 2·55-s + 4·57-s − 12·59-s − 14·61-s − 2·65-s − 6·67-s + 16·71-s − 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 1.07·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s − 1.56·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s + 1.37·53-s − 0.269·55-s + 0.529·57-s − 1.56·59-s − 1.79·61-s − 0.248·65-s − 0.733·67-s + 1.89·71-s − 0.234·73-s + ⋯

Functional equation

Λ(s)=(23520s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(23520s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2352023520    =    25⋅3⋅5⋅722^{5} \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 187.808187.808
Root analytic conductor: 13.704313.7043
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 23520, ( :1/2), −1)(2,\ 23520,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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−T 1 - T
5 1+T 1 + T
7 1 1
good11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p 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

−15.53970693391689, −15.25757611604552, −14.71976622953796, −13.97474631287626, −13.63873424594849, −13.23946147762369, −12.30379848633424, −11.99258201430211, −11.46647419092504, −10.80706819351282, −10.15524722716648, −9.667193088346547, −8.968468933262948, −8.548535709322328, −7.972265533062965, −7.376273369663436, −6.729671895061633, −6.263580429728765, −5.300883252792891, −4.801944563228680, −3.944693925627110, −3.427609066609248, −2.927804070368661, −1.793032963431239, −1.255768500814297, 0, 1.255768500814297, 1.793032963431239, 2.927804070368661, 3.427609066609248, 3.944693925627110, 4.801944563228680, 5.300883252792891, 6.263580429728765, 6.729671895061633, 7.376273369663436, 7.972265533062965, 8.548535709322328, 8.968468933262948, 9.667193088346547, 10.15524722716648, 10.80706819351282, 11.46647419092504, 11.99258201430211, 12.30379848633424, 13.23946147762369, 13.63873424594849, 13.97474631287626, 14.71976622953796, 15.25757611604552, 15.53970693391689

Graph of the ZZ-function along the critical line