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Properties

Label 2-35e2-1.1-c3-0-89
Degree 22
Conductor 12251225
Sign −1-1
Analytic cond. 72.277372.2773
Root an. cond. 8.501608.50160
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  + 2-s − 7·3-s − 7·4-s − 7·6-s − 15·8-s + 22·9-s − 43·11-s + 49·12-s + 28·13-s + 41·16-s − 91·17-s + 22·18-s + 35·19-s − 43·22-s + 162·23-s + 105·24-s + 28·26-s + 35·27-s + 160·29-s − 42·31-s + 161·32-s + 301·33-s − 91·34-s − 154·36-s − 314·37-s + 35·38-s − 196·39-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.34·3-s − 7/8·4-s − 0.476·6-s − 0.662·8-s + 0.814·9-s − 1.17·11-s + 1.17·12-s + 0.597·13-s + 0.640·16-s − 1.29·17-s + 0.288·18-s + 0.422·19-s − 0.416·22-s + 1.46·23-s + 0.893·24-s + 0.211·26-s + 0.249·27-s + 1.02·29-s − 0.243·31-s + 0.889·32-s + 1.58·33-s − 0.459·34-s − 0.712·36-s − 1.39·37-s + 0.149·38-s − 0.804·39-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 72.277372.2773
Root analytic conductor: 8.501608.50160
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1225, ( :3/2), −1)(2,\ 1225,\ (\ :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)
bad5 1 1
7 1 1
good2 1−T+p3T2 1 - T + p^{3} T^{2}
3 1+7T+p3T2 1 + 7 T + p^{3} T^{2}
11 1+43T+p3T2 1 + 43 T + p^{3} T^{2}
13 1−28T+p3T2 1 - 28 T + p^{3} T^{2}
17 1+91T+p3T2 1 + 91 T + p^{3} T^{2}
19 1−35T+p3T2 1 - 35 T + p^{3} T^{2}
23 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
29 1−160T+p3T2 1 - 160 T + p^{3} T^{2}
31 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
37 1+314T+p3T2 1 + 314 T + p^{3} T^{2}
41 1−203T+p3T2 1 - 203 T + p^{3} T^{2}
43 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
47 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
53 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
59 1−280T+p3T2 1 - 280 T + p^{3} T^{2}
61 1−518T+p3T2 1 - 518 T + p^{3} T^{2}
67 1−141T+p3T2 1 - 141 T + p^{3} T^{2}
71 1−412T+p3T2 1 - 412 T + p^{3} T^{2}
73 1−763T+p3T2 1 - 763 T + p^{3} T^{2}
79 1−510T+p3T2 1 - 510 T + p^{3} T^{2}
83 1+777T+p3T2 1 + 777 T + p^{3} T^{2}
89 1−945T+p3T2 1 - 945 T + p^{3} T^{2}
97 1+1246T+p3T2 1 + 1246 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

−8.914809895094261190331668071022, −8.246560856316936871129274975654, −7.00716148536209536910540526199, −6.26670559017336053728345099836, −5.16553865065355082646641963757, −5.09696743634974546613535499900, −3.93827318184214908039349194751, −2.72443318535649633635183444967, −0.940139905114092155886518375631, 0, 0.940139905114092155886518375631, 2.72443318535649633635183444967, 3.93827318184214908039349194751, 5.09696743634974546613535499900, 5.16553865065355082646641963757, 6.26670559017336053728345099836, 7.00716148536209536910540526199, 8.246560856316936871129274975654, 8.914809895094261190331668071022

Graph of the ZZ-function along the critical line