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Properties

Label 2-58800-1.1-c1-0-103
Degree 22
Conductor 5880058800
Sign −1-1
Analytic cond. 469.520469.520
Root an. cond. 21.668421.6684
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 + 9-s − 6·11-s − 13-s + 3·17-s − 4·19-s − 3·23-s − 27-s + 3·29-s + 5·31-s + 6·33-s + 10·37-s + 39-s − 9·41-s − 43-s − 3·51-s − 9·53-s + 4·57-s + 9·59-s − 11·61-s − 4·67-s + 3·69-s + 12·71-s − 10·73-s + 10·79-s + 81-s − 9·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.625·23-s − 0.192·27-s + 0.557·29-s + 0.898·31-s + 1.04·33-s + 1.64·37-s + 0.160·39-s − 1.40·41-s − 0.152·43-s − 0.420·51-s − 1.23·53-s + 0.529·57-s + 1.17·59-s − 1.40·61-s − 0.488·67-s + 0.361·69-s + 1.42·71-s − 1.17·73-s + 1.12·79-s + 1/9·81-s − 0.987·83-s + ⋯

Functional equation

Λ(s)=(58800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(58800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5880058800    =    24⋅3⋅52⋅722^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 469.520469.520
Root analytic conductor: 21.668421.6684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 58800, ( :1/2), −1)(2,\ 58800,\ (\ :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 1
7 1 1
good11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+11T+pT2 1 + 11 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−14.73002986022611, −13.96614208805759, −13.52564046452589, −12.98455798215748, −12.58105180403185, −12.12726465039134, −11.49959582172637, −11.03908445910568, −10.40988829594765, −10.05194826041469, −9.782560394805546, −8.784761774256016, −8.320122262185663, −7.650468643827094, −7.551878106521224, −6.472145799766786, −6.223492727452183, −5.551840393215168, −4.872167163134722, −4.679135654625971, −3.787783994139921, −3.001051566345191, −2.477158553851844, −1.755004674222598, −0.7469061219077777, 0, 0.7469061219077777, 1.755004674222598, 2.477158553851844, 3.001051566345191, 3.787783994139921, 4.679135654625971, 4.872167163134722, 5.551840393215168, 6.223492727452183, 6.472145799766786, 7.551878106521224, 7.650468643827094, 8.320122262185663, 8.784761774256016, 9.782560394805546, 10.05194826041469, 10.40988829594765, 11.03908445910568, 11.49959582172637, 12.12726465039134, 12.58105180403185, 12.98455798215748, 13.52564046452589, 13.96614208805759, 14.73002986022611

Graph of the ZZ-function along the critical line