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Properties

Label 2-208725-1.1-c1-0-27
Degree 22
Conductor 208725208725
Sign 11
Analytic cond. 1666.671666.67
Root an. cond. 40.824940.8249
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 7-s + 9-s − 2·12-s + 2·13-s + 4·16-s + 5·19-s + 21-s + 23-s + 27-s − 2·28-s − 8·29-s − 9·31-s − 2·36-s + 9·37-s + 2·39-s + 12·41-s + 4·43-s − 10·47-s + 4·48-s − 6·49-s − 4·52-s + 5·57-s + 12·59-s − 61-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 0.554·13-s + 16-s + 1.14·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s − 0.377·28-s − 1.48·29-s − 1.61·31-s − 1/3·36-s + 1.47·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s − 1.45·47-s + 0.577·48-s − 6/7·49-s − 0.554·52-s + 0.662·57-s + 1.56·59-s − 0.128·61-s + 0.125·63-s + ⋯

Functional equation

Λ(s)=(208725s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(208725s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 208725208725    =    3⋅52⋅112⋅233 \cdot 5^{2} \cdot 11^{2} \cdot 23
Sign: 11
Analytic conductor: 1666.671666.67
Root analytic conductor: 40.824940.8249
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 208725, ( :1/2), 1)(2,\ 208725,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2291696473.229169647
L(12)L(\frac12) ≈\approx 3.2291696473.229169647
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)
bad3 1−T 1 - T
5 1 1
11 1 1
23 1−T 1 - T
good2 1+pT2 1 + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+3T+pT2 1 + 3 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1−7T+pT2 1 - 7 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+T+pT2 1 + 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

−13.03143663237434, −12.90119877011867, −12.17645062688734, −11.55554332198250, −11.08301628111818, −10.80655593344666, −10.01839093743901, −9.517868476646001, −9.254013975375158, −8.971402287366922, −8.201148007526677, −7.795455286750519, −7.593599878550114, −6.916844176146069, −6.112571512594518, −5.731088099455178, −5.129627608179232, −4.749157530233660, −4.052281539232892, −3.580025035547990, −3.310912174928073, −2.383939351806920, −1.842297857673982, −1.060211048073994, −0.5708118324782255, 0.5708118324782255, 1.060211048073994, 1.842297857673982, 2.383939351806920, 3.310912174928073, 3.580025035547990, 4.052281539232892, 4.749157530233660, 5.129627608179232, 5.731088099455178, 6.112571512594518, 6.916844176146069, 7.593599878550114, 7.795455286750519, 8.201148007526677, 8.971402287366922, 9.254013975375158, 9.517868476646001, 10.01839093743901, 10.80655593344666, 11.08301628111818, 11.55554332198250, 12.17645062688734, 12.90119877011867, 13.03143663237434

Graph of the ZZ-function along the critical line