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Properties

Label 2-5733-1.1-c1-0-122
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  − 1.43·2-s + 0.0731·4-s + 0.926·5-s + 2.77·8-s − 1.33·10-s − 4.21·11-s − 13-s − 4.14·16-s − 2.87·17-s + 1.28·19-s + 0.0678·20-s + 6.06·22-s + 8.02·23-s − 4.14·25-s + 1.43·26-s − 3.28·29-s + 7.04·31-s + 0.413·32-s + 4.14·34-s + 8.57·37-s − 1.85·38-s + 2.57·40-s − 12.0·41-s + 7.14·43-s − 0.308·44-s − 11.5·46-s + 1.95·47-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.0365·4-s + 0.414·5-s + 0.980·8-s − 0.422·10-s − 1.27·11-s − 0.277·13-s − 1.03·16-s − 0.698·17-s + 0.295·19-s + 0.0151·20-s + 1.29·22-s + 1.67·23-s − 0.828·25-s + 0.282·26-s − 0.610·29-s + 1.26·31-s + 0.0731·32-s + 0.711·34-s + 1.40·37-s − 0.300·38-s + 0.406·40-s − 1.88·41-s + 1.08·43-s − 0.0464·44-s − 1.70·46-s + 0.284·47-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+1.43T+2T2 1 + 1.43T + 2T^{2}
5 1−0.926T+5T2 1 - 0.926T + 5T^{2}
11 1+4.21T+11T2 1 + 4.21T + 11T^{2}
17 1+2.87T+17T2 1 + 2.87T + 17T^{2}
19 1−1.28T+19T2 1 - 1.28T + 19T^{2}
23 1−8.02T+23T2 1 - 8.02T + 23T^{2}
29 1+3.28T+29T2 1 + 3.28T + 29T^{2}
31 1−7.04T+31T2 1 - 7.04T + 31T^{2}
37 1−8.57T+37T2 1 - 8.57T + 37T^{2}
41 1+12.0T+41T2 1 + 12.0T + 41T^{2}
43 1−7.14T+43T2 1 - 7.14T + 43T^{2}
47 1−1.95T+47T2 1 - 1.95T + 47T^{2}
53 1−5.14T+53T2 1 - 5.14T + 53T^{2}
59 1+7.33T+59T2 1 + 7.33T + 59T^{2}
61 1+7.75T+61T2 1 + 7.75T + 61T^{2}
67 1−12.0T+67T2 1 - 12.0T + 67T^{2}
71 1+10.7T+71T2 1 + 10.7T + 71T^{2}
73 1−8.32T+73T2 1 - 8.32T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1+3.80T+83T2 1 + 3.80T + 83T^{2}
89 1+5.64T+89T2 1 + 5.64T + 89T^{2}
97 1+6.90T+97T2 1 + 6.90T + 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

−7.88022923768521512617289112024, −7.30012484176485825628825041203, −6.54927108246297572888901124092, −5.53795868160919085390447662320, −4.93839525140265462414638901037, −4.20542055827670338463232149552, −2.93962049313617378947003947423, −2.21237405843254983204492030968, −1.11834739631990087472559495899, 0, 1.11834739631990087472559495899, 2.21237405843254983204492030968, 2.93962049313617378947003947423, 4.20542055827670338463232149552, 4.93839525140265462414638901037, 5.53795868160919085390447662320, 6.54927108246297572888901124092, 7.30012484176485825628825041203, 7.88022923768521512617289112024

Graph of the ZZ-function along the critical line