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Properties

Label 2-2850-1.1-c1-0-19
Degree 22
Conductor 28502850
Sign 11
Analytic cond. 22.757322.7573
Root an. cond. 4.770464.77046
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 + 8-s + 9-s + 4·11-s + 12-s − 4·14-s + 16-s + 2·17-s + 18-s + 19-s − 4·21-s + 4·22-s + 2·23-s + 24-s + 27-s − 4·28-s − 6·29-s + 6·31-s + 32-s + 4·33-s + 2·34-s + 36-s + 8·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-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 + 0.417·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 1/6·36-s + 1.31·37-s + ⋯

Functional equation

Λ(s)=(2850s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2850s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28502850    =    2⋅3⋅52⋅192 \cdot 3 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 22.757322.7573
Root analytic conductor: 4.770464.77046
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2850, ( :1/2), 1)(2,\ 2850,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4147980483.414798048
L(12)L(\frac12) ≈\approx 3.4147980483.414798048
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−T 1 - T
3 1−T 1 - T
5 1 1
19 1−T 1 - T
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 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 1−12T+pT2 1 - 12 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+2T+pT2 1 + 2 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

−9.075576842406541201241594770774, −7.80954843090816149956704777250, −7.22822931232102684143493050402, −6.24265288481434745554412487281, −6.04547848262920215925763806415, −4.68475865669594320244755229333, −3.85809577170289018122041589134, −3.25425866639495471915764029739, −2.44562386899404817546775712870, −1.04269711977505524501535875684, 1.04269711977505524501535875684, 2.44562386899404817546775712870, 3.25425866639495471915764029739, 3.85809577170289018122041589134, 4.68475865669594320244755229333, 6.04547848262920215925763806415, 6.24265288481434745554412487281, 7.22822931232102684143493050402, 7.80954843090816149956704777250, 9.075576842406541201241594770774

Graph of the ZZ-function along the critical line