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Properties

Label 2-920-1.1-c1-0-1
Degree 22
Conductor 920920
Sign 11
Analytic cond. 7.346237.34623
Root an. cond. 2.710392.71039
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  − 1.31·3-s − 5-s − 4.66·7-s − 1.28·9-s + 2.23·11-s − 2.80·13-s + 1.31·15-s + 7.63·17-s − 1.36·19-s + 6.11·21-s − 23-s + 25-s + 5.61·27-s + 8.94·29-s − 1.58·31-s − 2.92·33-s + 4.66·35-s − 1.40·37-s + 3.67·39-s + 10.7·41-s + 1.28·45-s − 7.26·47-s + 14.7·49-s − 10.0·51-s − 8.38·53-s − 2.23·55-s + 1.78·57-s + ⋯
L(s)  = 1  − 0.756·3-s − 0.447·5-s − 1.76·7-s − 0.427·9-s + 0.672·11-s − 0.776·13-s + 0.338·15-s + 1.85·17-s − 0.312·19-s + 1.33·21-s − 0.208·23-s + 0.200·25-s + 1.08·27-s + 1.66·29-s − 0.284·31-s − 0.508·33-s + 0.788·35-s − 0.230·37-s + 0.587·39-s + 1.67·41-s + 0.191·45-s − 1.05·47-s + 2.10·49-s − 1.40·51-s − 1.15·53-s − 0.300·55-s + 0.236·57-s + ⋯

Functional equation

Λ(s)=(920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 920920    =    23⋅5⋅232^{3} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 7.346237.34623
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 920, ( :1/2), 1)(2,\ 920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.71194983090.7119498309
L(12)L(\frac12) ≈\approx 0.71194983090.7119498309
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 1
5 1+T 1 + T
23 1+T 1 + T
good3 1+1.31T+3T2 1 + 1.31T + 3T^{2}
7 1+4.66T+7T2 1 + 4.66T + 7T^{2}
11 1−2.23T+11T2 1 - 2.23T + 11T^{2}
13 1+2.80T+13T2 1 + 2.80T + 13T^{2}
17 1−7.63T+17T2 1 - 7.63T + 17T^{2}
19 1+1.36T+19T2 1 + 1.36T + 19T^{2}
29 1−8.94T+29T2 1 - 8.94T + 29T^{2}
31 1+1.58T+31T2 1 + 1.58T + 31T^{2}
37 1+1.40T+37T2 1 + 1.40T + 37T^{2}
41 1−10.7T+41T2 1 - 10.7T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+7.26T+47T2 1 + 7.26T + 47T^{2}
53 1+8.38T+53T2 1 + 8.38T + 53T^{2}
59 1+4.88T+59T2 1 + 4.88T + 59T^{2}
61 1−4.33T+61T2 1 - 4.33T + 61T^{2}
67 1−8.54T+67T2 1 - 8.54T + 67T^{2}
71 1−8.81T+71T2 1 - 8.81T + 71T^{2}
73 1+5.26T+73T2 1 + 5.26T + 73T^{2}
79 1−7.08T+79T2 1 - 7.08T + 79T^{2}
83 1+4.59T+83T2 1 + 4.59T + 83T^{2}
89 1+4.70T+89T2 1 + 4.70T + 89T^{2}
97 1−16.5T+97T2 1 - 16.5T + 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

−9.981241531834909262063454138761, −9.519251041317907398170818991044, −8.416762214690052323266687001298, −7.38411124307599858031450644681, −6.47179509615879399356381659089, −5.95322277109494328558338176585, −4.87667813490802120246453902636, −3.61468483386655638528982792785, −2.85040023823048831759740890976, −0.67265184045756509530387203496, 0.67265184045756509530387203496, 2.85040023823048831759740890976, 3.61468483386655638528982792785, 4.87667813490802120246453902636, 5.95322277109494328558338176585, 6.47179509615879399356381659089, 7.38411124307599858031450644681, 8.416762214690052323266687001298, 9.519251041317907398170818991044, 9.981241531834909262063454138761

Graph of the ZZ-function along the critical line