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Properties

Label 2-2850-1.1-c1-0-26
Degree 22
Conductor 28502850
Sign 11
Analytic cond. 22.757322.7573
Root an. cond. 4.770464.77046
Motivic weight 11
Arithmetic yes
Rational no
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 + 2.73·7-s + 8-s + 9-s + 5.19·11-s − 12-s + 4.73·13-s + 2.73·14-s + 16-s − 2.73·17-s + 18-s + 19-s − 2.73·21-s + 5.19·22-s + 8.46·23-s − 24-s + 4.73·26-s − 27-s + 2.73·28-s − 9.19·29-s − 5.92·31-s + 32-s − 5.19·33-s − 2.73·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.03·7-s + 0.353·8-s + 0.333·9-s + 1.56·11-s − 0.288·12-s + 1.31·13-s + 0.730·14-s + 0.250·16-s − 0.662·17-s + 0.235·18-s + 0.229·19-s − 0.596·21-s + 1.10·22-s + 1.76·23-s − 0.204·24-s + 0.928·26-s − 0.192·27-s + 0.516·28-s − 1.70·29-s − 1.06·31-s + 0.176·32-s − 0.904·33-s − 0.468·34-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: no
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 3.2526201283.252620128
L(12)L(\frac12) ≈\approx 3.2526201283.252620128
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 1−2.73T+7T2 1 - 2.73T + 7T^{2}
11 1−5.19T+11T2 1 - 5.19T + 11T^{2}
13 1−4.73T+13T2 1 - 4.73T + 13T^{2}
17 1+2.73T+17T2 1 + 2.73T + 17T^{2}
23 1−8.46T+23T2 1 - 8.46T + 23T^{2}
29 1+9.19T+29T2 1 + 9.19T + 29T^{2}
31 1+5.92T+31T2 1 + 5.92T + 31T^{2}
37 1+8.92T+37T2 1 + 8.92T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−8.73T+43T2 1 - 8.73T + 43T^{2}
47 1+3.46T+47T2 1 + 3.46T + 47T^{2}
53 1+4.66T+53T2 1 + 4.66T + 53T^{2}
59 1−2.19T+59T2 1 - 2.19T + 59T^{2}
61 1−8.26T+61T2 1 - 8.26T + 61T^{2}
67 1+5.19T+67T2 1 + 5.19T + 67T^{2}
71 1+10.1T+71T2 1 + 10.1T + 71T^{2}
73 1−14.4T+73T2 1 - 14.4T + 73T^{2}
79 1−5.92T+79T2 1 - 5.92T + 79T^{2}
83 1−16.6T+83T2 1 - 16.6T + 83T^{2}
89 1−3.92T+89T2 1 - 3.92T + 89T^{2}
97 1−17.1T+97T2 1 - 17.1T + 97T^{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

−8.983393638329352776948390823111, −7.88259923770659831535443333040, −6.99688805945246289905831484142, −6.48653844734527045112358021653, −5.57697346970628539897524518216, −4.97491738302514296442542307338, −4.02340333180061077855155384203, −3.49628878613053952275137333029, −1.88897408435967492974056121666, −1.17707155491389899811605176996, 1.17707155491389899811605176996, 1.88897408435967492974056121666, 3.49628878613053952275137333029, 4.02340333180061077855155384203, 4.97491738302514296442542307338, 5.57697346970628539897524518216, 6.48653844734527045112358021653, 6.99688805945246289905831484142, 7.88259923770659831535443333040, 8.983393638329352776948390823111

Graph of the ZZ-function along the critical line