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Properties

Label 2-4600-1.1-c1-0-46
Degree 22
Conductor 46004600
Sign −1-1
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  − 1.78·3-s − 1.75·7-s + 0.192·9-s − 4.77·11-s − 1.72·13-s + 7.81·17-s − 2.43·19-s + 3.13·21-s + 23-s + 5.01·27-s + 7.86·29-s + 6.14·31-s + 8.53·33-s + 6.83·37-s + 3.09·39-s − 2.50·41-s + 3.26·43-s − 8.46·47-s − 3.91·49-s − 13.9·51-s + 2.76·53-s + 4.35·57-s + 1.91·59-s − 3.50·61-s − 0.337·63-s + 12.7·67-s − 1.78·69-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.663·7-s + 0.0640·9-s − 1.43·11-s − 0.479·13-s + 1.89·17-s − 0.559·19-s + 0.684·21-s + 0.208·23-s + 0.965·27-s + 1.46·29-s + 1.10·31-s + 1.48·33-s + 1.12·37-s + 0.494·39-s − 0.391·41-s + 0.498·43-s − 1.23·47-s − 0.559·49-s − 1.95·51-s + 0.379·53-s + 0.577·57-s + 0.249·59-s − 0.448·61-s − 0.0424·63-s + 1.55·67-s − 0.215·69-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: −1-1
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4600, ( :1/2), −1)(2,\ 4600,\ (\ :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)
bad2 1 1
5 1 1
23 1−T 1 - T
good3 1+1.78T+3T2 1 + 1.78T + 3T^{2}
7 1+1.75T+7T2 1 + 1.75T + 7T^{2}
11 1+4.77T+11T2 1 + 4.77T + 11T^{2}
13 1+1.72T+13T2 1 + 1.72T + 13T^{2}
17 1−7.81T+17T2 1 - 7.81T + 17T^{2}
19 1+2.43T+19T2 1 + 2.43T + 19T^{2}
29 1−7.86T+29T2 1 - 7.86T + 29T^{2}
31 1−6.14T+31T2 1 - 6.14T + 31T^{2}
37 1−6.83T+37T2 1 - 6.83T + 37T^{2}
41 1+2.50T+41T2 1 + 2.50T + 41T^{2}
43 1−3.26T+43T2 1 - 3.26T + 43T^{2}
47 1+8.46T+47T2 1 + 8.46T + 47T^{2}
53 1−2.76T+53T2 1 - 2.76T + 53T^{2}
59 1−1.91T+59T2 1 - 1.91T + 59T^{2}
61 1+3.50T+61T2 1 + 3.50T + 61T^{2}
67 1−12.7T+67T2 1 - 12.7T + 67T^{2}
71 1+13.3T+71T2 1 + 13.3T + 71T^{2}
73 1+0.0111T+73T2 1 + 0.0111T + 73T^{2}
79 1+16.6T+79T2 1 + 16.6T + 79T^{2}
83 1−2.64T+83T2 1 - 2.64T + 83T^{2}
89 1+13.1T+89T2 1 + 13.1T + 89T^{2}
97 1+15.8T+97T2 1 + 15.8T + 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

−8.001104239226936589353669763652, −7.13556071001258306351585766680, −6.35174500800127661412452151362, −5.75355989184073701771855582160, −5.12681855701928224377247406110, −4.46259266896788335813175768766, −3.11527968560304055457293879070, −2.67019427601249981273358556028, −1.04469740306387750651871938139, 0, 1.04469740306387750651871938139, 2.67019427601249981273358556028, 3.11527968560304055457293879070, 4.46259266896788335813175768766, 5.12681855701928224377247406110, 5.75355989184073701771855582160, 6.35174500800127661412452151362, 7.13556071001258306351585766680, 8.001104239226936589353669763652

Graph of the ZZ-function along the critical line