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Properties

Label 2-1425-1.1-c1-0-6
Degree 22
Conductor 14251425
Sign 11
Analytic cond. 11.378611.3786
Root an. cond. 3.373233.37323
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  − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s + 4·11-s − 12-s − 2·13-s + 4·14-s − 16-s − 2·17-s − 18-s − 19-s − 4·21-s − 4·22-s + 4·23-s + 3·24-s + 2·26-s + 27-s + 4·28-s − 2·29-s − 5·32-s + 4·33-s + 2·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.872·21-s − 0.852·22-s + 0.834·23-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.883·32-s + 0.696·33-s + 0.342·34-s − 1/6·36-s + ⋯

Functional equation

Λ(s)=(1425s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1425s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14251425    =    3⋅52⋅193 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 11.378611.3786
Root analytic conductor: 3.373233.37323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1425, ( :1/2), 1)(2,\ 1425,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.97006510330.9700651033
L(12)L(\frac12) ≈\approx 0.97006510330.9700651033
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
19 1+T 1 + T
good2 1+T+pT2 1 + T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−9.456957912327740145299523356238, −8.942236798090976512890571214951, −8.204261797095146615633952733135, −7.02357868670760652523158856931, −6.70552074711622254156145782215, −5.38607911495288050360483967057, −4.17893424077245633552095753386, −3.54490000396289616314690825839, −2.29629701215321469443462239002, −0.76854595244344636007624940810, 0.76854595244344636007624940810, 2.29629701215321469443462239002, 3.54490000396289616314690825839, 4.17893424077245633552095753386, 5.38607911495288050360483967057, 6.70552074711622254156145782215, 7.02357868670760652523158856931, 8.204261797095146615633952733135, 8.942236798090976512890571214951, 9.456957912327740145299523356238

Graph of the ZZ-function along the critical line