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Properties

Label 2-92e2-1.1-c1-0-53
Degree 22
Conductor 84648464
Sign 11
Analytic cond. 67.585367.5853
Root an. cond. 8.221038.22103
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  − 2.44·3-s − 3·5-s + 0.449·7-s + 2.99·9-s + 2.44·11-s + 5.89·13-s + 7.34·15-s − 4.89·17-s − 3.55·19-s − 1.10·21-s + 4·25-s + 7.89·29-s + 10.8·31-s − 5.99·33-s − 1.34·35-s + 8·37-s − 14.4·39-s + 7.89·41-s − 6.89·43-s − 8.99·45-s − 1.55·47-s − 6.79·49-s + 11.9·51-s + 8.79·53-s − 7.34·55-s + 8.69·57-s − 5.34·59-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.34·5-s + 0.169·7-s + 0.999·9-s + 0.738·11-s + 1.63·13-s + 1.89·15-s − 1.18·17-s − 0.814·19-s − 0.240·21-s + 0.800·25-s + 1.46·29-s + 1.95·31-s − 1.04·33-s − 0.227·35-s + 1.31·37-s − 2.31·39-s + 1.23·41-s − 1.05·43-s − 1.34·45-s − 0.226·47-s − 0.971·49-s + 1.68·51-s + 1.20·53-s − 0.990·55-s + 1.15·57-s − 0.696·59-s + ⋯

Functional equation

Λ(s)=(8464s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8464s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 84648464    =    24⋅2322^{4} \cdot 23^{2}
Sign: 11
Analytic conductor: 67.585367.5853
Root analytic conductor: 8.221038.22103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8464, ( :1/2), 1)(2,\ 8464,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.89318843690.8931884369
L(12)L(\frac12) ≈\approx 0.89318843690.8931884369
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
23 1 1
good3 1+2.44T+3T2 1 + 2.44T + 3T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
7 1−0.449T+7T2 1 - 0.449T + 7T^{2}
11 1−2.44T+11T2 1 - 2.44T + 11T^{2}
13 1−5.89T+13T2 1 - 5.89T + 13T^{2}
17 1+4.89T+17T2 1 + 4.89T + 17T^{2}
19 1+3.55T+19T2 1 + 3.55T + 19T^{2}
29 1−7.89T+29T2 1 - 7.89T + 29T^{2}
31 1−10.8T+31T2 1 - 10.8T + 31T^{2}
37 1−8T+37T2 1 - 8T + 37T^{2}
41 1−7.89T+41T2 1 - 7.89T + 41T^{2}
43 1+6.89T+43T2 1 + 6.89T + 43T^{2}
47 1+1.55T+47T2 1 + 1.55T + 47T^{2}
53 1−8.79T+53T2 1 - 8.79T + 53T^{2}
59 1+5.34T+59T2 1 + 5.34T + 59T^{2}
61 1−5.89T+61T2 1 - 5.89T + 61T^{2}
67 1+13.7T+67T2 1 + 13.7T + 67T^{2}
71 1−4.44T+71T2 1 - 4.44T + 71T^{2}
73 1+1.89T+73T2 1 + 1.89T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+6.89T+83T2 1 + 6.89T + 83T^{2}
89 1−8.79T+89T2 1 - 8.79T + 89T^{2}
97 1+1.89T+97T2 1 + 1.89T + 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

−7.78098523552231746436921124402, −6.83541994547435757093443458090, −6.29680431745563089588601182998, −6.05573371932215340017507843255, −4.69976944413607947119600354203, −4.46131312179153461526264297037, −3.80485494160127383364013876299, −2.75243464786618541414837244980, −1.29578320510573281359593748324, −0.57114632521050694568744861385, 0.57114632521050694568744861385, 1.29578320510573281359593748324, 2.75243464786618541414837244980, 3.80485494160127383364013876299, 4.46131312179153461526264297037, 4.69976944413607947119600354203, 6.05573371932215340017507843255, 6.29680431745563089588601182998, 6.83541994547435757093443458090, 7.78098523552231746436921124402

Graph of the ZZ-function along the critical line