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Properties

Label 2-920-1.1-c1-0-11
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  + 3.30·3-s − 5-s − 2.55·7-s + 7.93·9-s − 2.72·11-s + 7.12·13-s − 3.30·15-s + 0.924·17-s + 7.51·19-s − 8.43·21-s − 23-s + 25-s + 16.3·27-s − 2.38·29-s + 0.866·31-s − 9.00·33-s + 2.55·35-s + 0.352·37-s + 23.5·39-s + 4.34·41-s − 7.93·45-s − 13.3·47-s − 0.495·49-s + 3.05·51-s + 3.99·53-s + 2.72·55-s + 24.8·57-s + ⋯
L(s)  = 1  + 1.90·3-s − 0.447·5-s − 0.963·7-s + 2.64·9-s − 0.821·11-s + 1.97·13-s − 0.853·15-s + 0.224·17-s + 1.72·19-s − 1.84·21-s − 0.208·23-s + 0.200·25-s + 3.13·27-s − 0.442·29-s + 0.155·31-s − 1.56·33-s + 0.431·35-s + 0.0580·37-s + 3.77·39-s + 0.677·41-s − 1.18·45-s − 1.94·47-s − 0.0707·49-s + 0.427·51-s + 0.548·53-s + 0.367·55-s + 3.29·57-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 2.7860715652.786071565
L(12)L(\frac12) ≈\approx 2.7860715652.786071565
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 1−3.30T+3T2 1 - 3.30T + 3T^{2}
7 1+2.55T+7T2 1 + 2.55T + 7T^{2}
11 1+2.72T+11T2 1 + 2.72T + 11T^{2}
13 1−7.12T+13T2 1 - 7.12T + 13T^{2}
17 1−0.924T+17T2 1 - 0.924T + 17T^{2}
19 1−7.51T+19T2 1 - 7.51T + 19T^{2}
29 1+2.38T+29T2 1 + 2.38T + 29T^{2}
31 1−0.866T+31T2 1 - 0.866T + 31T^{2}
37 1−0.352T+37T2 1 - 0.352T + 37T^{2}
41 1−4.34T+41T2 1 - 4.34T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+13.3T+47T2 1 + 13.3T + 47T^{2}
53 1−3.99T+53T2 1 - 3.99T + 53T^{2}
59 1+3.84T+59T2 1 + 3.84T + 59T^{2}
61 1+9.14T+61T2 1 + 9.14T + 61T^{2}
67 1+3.15T+67T2 1 + 3.15T + 67T^{2}
71 1+6.07T+71T2 1 + 6.07T + 71T^{2}
73 1+11.3T+73T2 1 + 11.3T + 73T^{2}
79 1+12.0T+79T2 1 + 12.0T + 79T^{2}
83 1+6.35T+83T2 1 + 6.35T + 83T^{2}
89 1+9.71T+89T2 1 + 9.71T + 89T^{2}
97 1−8.76T+97T2 1 - 8.76T + 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

−9.807896551322860008838534651220, −9.181268409898647740169010277397, −8.381310241584692499435959626850, −7.78039899087033012236440891210, −7.00007863474549330956034237589, −5.84438893151810673248538426668, −4.32983811528020909020172018173, −3.29091365915199499416758983718, −3.07269763056208165100716382094, −1.43596666248003605840752136528, 1.43596666248003605840752136528, 3.07269763056208165100716382094, 3.29091365915199499416758983718, 4.32983811528020909020172018173, 5.84438893151810673248538426668, 7.00007863474549330956034237589, 7.78039899087033012236440891210, 8.381310241584692499435959626850, 9.181268409898647740169010277397, 9.807896551322860008838534651220

Graph of the ZZ-function along the critical line