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Properties

Label 2-357-1.1-c1-0-1
Degree 22
Conductor 357357
Sign 11
Analytic cond. 2.850652.85065
Root an. cond. 1.688381.68838
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  − 2.06·2-s − 3-s + 2.27·4-s − 0.238·5-s + 2.06·6-s + 7-s − 0.558·8-s + 9-s + 0.491·10-s − 5.40·11-s − 2.27·12-s + 0.238·13-s − 2.06·14-s + 0.238·15-s − 3.38·16-s − 17-s − 2.06·18-s + 7.89·19-s − 0.540·20-s − 21-s + 11.1·22-s + 6.38·23-s + 0.558·24-s − 4.94·25-s − 0.491·26-s − 27-s + 2.27·28-s + ⋯
L(s)  = 1  − 1.46·2-s − 0.577·3-s + 1.13·4-s − 0.106·5-s + 0.843·6-s + 0.377·7-s − 0.197·8-s + 0.333·9-s + 0.155·10-s − 1.62·11-s − 0.655·12-s + 0.0660·13-s − 0.552·14-s + 0.0614·15-s − 0.846·16-s − 0.242·17-s − 0.487·18-s + 1.81·19-s − 0.120·20-s − 0.218·21-s + 2.38·22-s + 1.33·23-s + 0.113·24-s − 0.988·25-s − 0.0964·26-s − 0.192·27-s + 0.429·28-s + ⋯

Functional equation

Λ(s)=(357s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(357s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 357357    =    3⋅7⋅173 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 2.850652.85065
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 357, ( :1/2), 1)(2,\ 357,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.49903631480.4990363148
L(12)L(\frac12) ≈\approx 0.49903631480.4990363148
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+T 1 + T
7 1−T 1 - T
17 1+T 1 + T
good2 1+2.06T+2T2 1 + 2.06T + 2T^{2}
5 1+0.238T+5T2 1 + 0.238T + 5T^{2}
11 1+5.40T+11T2 1 + 5.40T + 11T^{2}
13 1−0.238T+13T2 1 - 0.238T + 13T^{2}
19 1−7.89T+19T2 1 - 7.89T + 19T^{2}
23 1−6.38T+23T2 1 - 6.38T + 23T^{2}
29 1−4.13T+29T2 1 - 4.13T + 29T^{2}
31 1−6.18T+31T2 1 - 6.18T + 31T^{2}
37 1−5.64T+37T2 1 - 5.64T + 37T^{2}
41 1+6.30T+41T2 1 + 6.30T + 41T^{2}
43 1−10.8T+43T2 1 - 10.8T + 43T^{2}
47 1−6.18T+47T2 1 - 6.18T + 47T^{2}
53 1−12.1T+53T2 1 - 12.1T + 53T^{2}
59 1+4.14T+59T2 1 + 4.14T + 59T^{2}
61 1+2.55T+61T2 1 + 2.55T + 61T^{2}
67 1−5.45T+67T2 1 - 5.45T + 67T^{2}
71 1−9.72T+71T2 1 - 9.72T + 71T^{2}
73 1+14.9T+73T2 1 + 14.9T + 73T^{2}
79 1+16.8T+79T2 1 + 16.8T + 79T^{2}
83 1−5.45T+83T2 1 - 5.45T + 83T^{2}
89 1+6.47T+89T2 1 + 6.47T + 89T^{2}
97 1+2.06T+97T2 1 + 2.06T + 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

−11.18183079890896129994370095848, −10.39776777972413569729574069695, −9.738720151306175736757794837053, −8.660182292476093748064227815120, −7.73047959398978833843427895427, −7.18032724620857925640044392391, −5.68641473272541393924609196502, −4.67942543603881613674176278989, −2.65319336574567211412948083003, −0.903628635718627513723344422918, 0.903628635718627513723344422918, 2.65319336574567211412948083003, 4.67942543603881613674176278989, 5.68641473272541393924609196502, 7.18032724620857925640044392391, 7.73047959398978833843427895427, 8.660182292476093748064227815120, 9.738720151306175736757794837053, 10.39776777972413569729574069695, 11.18183079890896129994370095848

Graph of the ZZ-function along the critical line