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Properties

Label 2-5733-1.1-c1-0-77
Degree 22
Conductor 57335733
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.17·2-s − 0.616·4-s + 3.14·5-s − 3.07·8-s + 3.70·10-s + 0.773·11-s − 13-s − 2.38·16-s − 5.75·17-s + 1.22·19-s − 1.94·20-s + 0.909·22-s + 2.99·23-s + 4.91·25-s − 1.17·26-s + 2.46·29-s + 6.13·31-s + 3.34·32-s − 6.76·34-s + 4.99·37-s + 1.43·38-s − 9.69·40-s − 2.55·41-s − 2.73·43-s − 0.476·44-s + 3.51·46-s + 5.37·47-s + ⋯
L(s)  = 1  + 0.831·2-s − 0.308·4-s + 1.40·5-s − 1.08·8-s + 1.17·10-s + 0.233·11-s − 0.277·13-s − 0.596·16-s − 1.39·17-s + 0.280·19-s − 0.434·20-s + 0.193·22-s + 0.623·23-s + 0.982·25-s − 0.230·26-s + 0.458·29-s + 1.10·31-s + 0.591·32-s − 1.16·34-s + 0.821·37-s + 0.233·38-s − 1.53·40-s − 0.399·41-s − 0.416·43-s − 0.0718·44-s + 0.518·46-s + 0.783·47-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: 11
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: 00
Selberg data: (2, 5733, ( :1/2), 1)(2,\ 5733,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2652836553.265283655
L(12)L(\frac12) ≈\approx 3.2652836553.265283655
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 1−1.17T+2T2 1 - 1.17T + 2T^{2}
5 1−3.14T+5T2 1 - 3.14T + 5T^{2}
11 1−0.773T+11T2 1 - 0.773T + 11T^{2}
17 1+5.75T+17T2 1 + 5.75T + 17T^{2}
19 1−1.22T+19T2 1 - 1.22T + 19T^{2}
23 1−2.99T+23T2 1 - 2.99T + 23T^{2}
29 1−2.46T+29T2 1 - 2.46T + 29T^{2}
31 1−6.13T+31T2 1 - 6.13T + 31T^{2}
37 1−4.99T+37T2 1 - 4.99T + 37T^{2}
41 1+2.55T+41T2 1 + 2.55T + 41T^{2}
43 1+2.73T+43T2 1 + 2.73T + 43T^{2}
47 1−5.37T+47T2 1 - 5.37T + 47T^{2}
53 1−9.79T+53T2 1 - 9.79T + 53T^{2}
59 1−2.50T+59T2 1 - 2.50T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1−4.32T+67T2 1 - 4.32T + 67T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1−5.17T+73T2 1 - 5.17T + 73T^{2}
79 1+0.542T+79T2 1 + 0.542T + 79T^{2}
83 1−15.2T+83T2 1 - 15.2T + 83T^{2}
89 1−9.23T+89T2 1 - 9.23T + 89T^{2}
97 1−1.26T+97T2 1 - 1.26T + 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

−8.330781053407755540629480184515, −7.08266807768052354606464084676, −6.45767633932950488412713987783, −5.91988538438232919854733187587, −5.14650410777053655462709631322, −4.64550056822500227159722985399, −3.79781777032719919896746753269, −2.73803075443526072774414710598, −2.18420729495515010390116591407, −0.851841989603352015285704727387, 0.851841989603352015285704727387, 2.18420729495515010390116591407, 2.73803075443526072774414710598, 3.79781777032719919896746753269, 4.64550056822500227159722985399, 5.14650410777053655462709631322, 5.91988538438232919854733187587, 6.45767633932950488412713987783, 7.08266807768052354606464084676, 8.330781053407755540629480184515

Graph of the ZZ-function along the critical line