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Properties

Label 2-5712-1.1-c1-0-33
Degree 22
Conductor 57125712
Sign 11
Analytic cond. 45.610545.6105
Root an. cond. 6.753556.75355
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-s + 0.414·5-s + 7-s + 9-s − 11-s − 1.58·13-s + 0.414·15-s − 17-s + 3.58·19-s + 21-s + 3.82·23-s − 4.82·25-s + 27-s + 6.82·29-s + 9.89·31-s − 33-s + 0.414·35-s − 8.24·37-s − 1.58·39-s − 4.07·41-s + 0.171·43-s + 0.414·45-s + 2.58·47-s + 49-s − 51-s − 2.24·53-s − 0.414·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.185·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s − 0.439·13-s + 0.106·15-s − 0.242·17-s + 0.822·19-s + 0.218·21-s + 0.798·23-s − 0.965·25-s + 0.192·27-s + 1.26·29-s + 1.77·31-s − 0.174·33-s + 0.0700·35-s − 1.35·37-s − 0.253·39-s − 0.635·41-s + 0.0261·43-s + 0.0617·45-s + 0.377·47-s + 0.142·49-s − 0.140·51-s − 0.308·53-s − 0.0558·55-s + ⋯

Functional equation

Λ(s)=(5712s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5712s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57125712    =    24⋅3⋅7⋅172^{4} \cdot 3 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 45.610545.6105
Root analytic conductor: 6.753556.75355
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5712, ( :1/2), 1)(2,\ 5712,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7335610182.733561018
L(12)L(\frac12) ≈\approx 2.7335610182.733561018
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
3 1−T 1 - T
7 1−T 1 - T
17 1+T 1 + T
good5 1−0.414T+5T2 1 - 0.414T + 5T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1+1.58T+13T2 1 + 1.58T + 13T^{2}
19 1−3.58T+19T2 1 - 3.58T + 19T^{2}
23 1−3.82T+23T2 1 - 3.82T + 23T^{2}
29 1−6.82T+29T2 1 - 6.82T + 29T^{2}
31 1−9.89T+31T2 1 - 9.89T + 31T^{2}
37 1+8.24T+37T2 1 + 8.24T + 37T^{2}
41 1+4.07T+41T2 1 + 4.07T + 41T^{2}
43 1−0.171T+43T2 1 - 0.171T + 43T^{2}
47 1−2.58T+47T2 1 - 2.58T + 47T^{2}
53 1+2.24T+53T2 1 + 2.24T + 53T^{2}
59 1−3.75T+59T2 1 - 3.75T + 59T^{2}
61 1−10.7T+61T2 1 - 10.7T + 61T^{2}
67 1−7.65T+67T2 1 - 7.65T + 67T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1+3.17T+73T2 1 + 3.17T + 73T^{2}
79 1+10.7T+79T2 1 + 10.7T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−10.8T+89T2 1 - 10.8T + 89T^{2}
97 1−1.65T+97T2 1 - 1.65T + 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

−8.347800100291707938316455271083, −7.38449536879295335366965804200, −6.89420283550869055549455107642, −5.98437824275424349483824455126, −5.07670968508387231612777342469, −4.58951721246886266343158634929, −3.54017576023724474879045079164, −2.77937453064625149295467605284, −1.98438468724894009061767019571, −0.870825852437118525615580797732, 0.870825852437118525615580797732, 1.98438468724894009061767019571, 2.77937453064625149295467605284, 3.54017576023724474879045079164, 4.58951721246886266343158634929, 5.07670968508387231612777342469, 5.98437824275424349483824455126, 6.89420283550869055549455107642, 7.38449536879295335366965804200, 8.347800100291707938316455271083

Graph of the ZZ-function along the critical line