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Properties

Label 2-5733-1.1-c1-0-90
Degree 22
Conductor 57335733
Sign −1-1
Analytic cond. 45.778245.7782
Root an. cond. 6.765966.76596
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 4.81·4-s − 3.81·5-s − 7.34·8-s + 9.95·10-s + 4.73·11-s − 13-s + 9.55·16-s − 5.22·17-s − 2.92·19-s − 18.3·20-s − 12.3·22-s − 3.33·23-s + 9.55·25-s + 2.61·26-s + 0.922·29-s + 7.51·31-s − 10.2·32-s + 13.6·34-s + 0.154·37-s + 7.62·38-s + 28.0·40-s + 6.36·41-s − 6.55·43-s + 22.8·44-s + 8.69·46-s + 9.03·47-s + ⋯
L(s)  = 1  − 1.84·2-s + 2.40·4-s − 1.70·5-s − 2.59·8-s + 3.14·10-s + 1.42·11-s − 0.277·13-s + 2.38·16-s − 1.26·17-s − 0.670·19-s − 4.10·20-s − 2.63·22-s − 0.694·23-s + 1.91·25-s + 0.511·26-s + 0.171·29-s + 1.35·31-s − 1.81·32-s + 2.33·34-s + 0.0254·37-s + 1.23·38-s + 4.43·40-s + 0.994·41-s − 0.999·43-s + 3.43·44-s + 1.28·46-s + 1.31·47-s + ⋯

Functional equation

Λ(s)=(5733s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5733s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57335733    =    32⋅72⋅133^{2} \cdot 7^{2} \cdot 13
Sign: −1-1
Analytic conductor: 45.778245.7782
Root analytic conductor: 6.765966.76596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5733, ( :1/2), −1)(2,\ 5733,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
7 1 1
13 1+T 1 + T
good2 1+2.61T+2T2 1 + 2.61T + 2T^{2}
5 1+3.81T+5T2 1 + 3.81T + 5T^{2}
11 1−4.73T+11T2 1 - 4.73T + 11T^{2}
17 1+5.22T+17T2 1 + 5.22T + 17T^{2}
19 1+2.92T+19T2 1 + 2.92T + 19T^{2}
23 1+3.33T+23T2 1 + 3.33T + 23T^{2}
29 1−0.922T+29T2 1 - 0.922T + 29T^{2}
31 1−7.51T+31T2 1 - 7.51T + 31T^{2}
37 1−0.154T+37T2 1 - 0.154T + 37T^{2}
41 1−6.36T+41T2 1 - 6.36T + 41T^{2}
43 1+6.55T+43T2 1 + 6.55T + 43T^{2}
47 1−9.03T+47T2 1 - 9.03T + 47T^{2}
53 1+8.55T+53T2 1 + 8.55T + 53T^{2}
59 1−3.95T+59T2 1 - 3.95T + 59T^{2}
61 1+12.4T+61T2 1 + 12.4T + 61T^{2}
67 1+10.6T+67T2 1 + 10.6T + 67T^{2}
71 1−6.58T+71T2 1 - 6.58T + 71T^{2}
73 1−7.73T+73T2 1 - 7.73T + 73T^{2}
79 1−13.3T+79T2 1 - 13.3T + 79T^{2}
83 1+1.40T+83T2 1 + 1.40T + 83T^{2}
89 1+1.96T+89T2 1 + 1.96T + 89T^{2}
97 1−2.11T+97T2 1 - 2.11T + 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

−8.018418909771570376364735413933, −7.28129006344996689288496539812, −6.65051742935430940916553814124, −6.22990041440798402899152382151, −4.56729295247629563909694239013, −4.02809745942750714628333507919, −3.01951775836285878222624563232, −2.02128732798285343792710446009, −0.917038887603706134983942892517, 0, 0.917038887603706134983942892517, 2.02128732798285343792710446009, 3.01951775836285878222624563232, 4.02809745942750714628333507919, 4.56729295247629563909694239013, 6.22990041440798402899152382151, 6.65051742935430940916553814124, 7.28129006344996689288496539812, 8.018418909771570376364735413933

Graph of the ZZ-function along the critical line