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Properties

Label 2-399-1.1-c1-0-12
Degree 22
Conductor 399399
Sign 11
Analytic cond. 3.186033.18603
Root an. cond. 1.784941.78494
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  + 1.39·2-s + 3-s − 0.0513·4-s + 3.05·5-s + 1.39·6-s − 7-s − 2.86·8-s + 9-s + 4.25·10-s + 5.31·11-s − 0.0513·12-s − 3.31·13-s − 1.39·14-s + 3.05·15-s − 3.89·16-s + 0.948·17-s + 1.39·18-s − 19-s − 0.156·20-s − 21-s + 7.41·22-s + 1.31·23-s − 2.86·24-s + 4.31·25-s − 4.62·26-s + 27-s + 0.0513·28-s + ⋯
L(s)  = 1  + 0.987·2-s + 0.577·3-s − 0.0256·4-s + 1.36·5-s + 0.569·6-s − 0.377·7-s − 1.01·8-s + 0.333·9-s + 1.34·10-s + 1.60·11-s − 0.0148·12-s − 0.918·13-s − 0.373·14-s + 0.787·15-s − 0.973·16-s + 0.230·17-s + 0.329·18-s − 0.229·19-s − 0.0350·20-s − 0.218·21-s + 1.58·22-s + 0.273·23-s − 0.584·24-s + 0.862·25-s − 0.906·26-s + 0.192·27-s + 0.00970·28-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 3.186033.18603
Root analytic conductor: 1.784941.78494
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 399, ( :1/2), 1)(2,\ 399,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7615066072.761506607
L(12)L(\frac12) ≈\approx 2.7615066072.761506607
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
19 1+T 1 + T
good2 1−1.39T+2T2 1 - 1.39T + 2T^{2}
5 1−3.05T+5T2 1 - 3.05T + 5T^{2}
11 1−5.31T+11T2 1 - 5.31T + 11T^{2}
13 1+3.31T+13T2 1 + 3.31T + 13T^{2}
17 1−0.948T+17T2 1 - 0.948T + 17T^{2}
23 1−1.31T+23T2 1 - 1.31T + 23T^{2}
29 1+7.84T+29T2 1 + 7.84T + 29T^{2}
31 1+4.79T+31T2 1 + 4.79T + 31T^{2}
37 1+8.62T+37T2 1 + 8.62T + 37T^{2}
41 1−11.3T+41T2 1 - 11.3T + 41T^{2}
43 1−3.20T+43T2 1 - 3.20T + 43T^{2}
47 1+5.84T+47T2 1 + 5.84T + 47T^{2}
53 1+2.77T+53T2 1 + 2.77T + 53T^{2}
59 1+8.20T+59T2 1 + 8.20T + 59T^{2}
61 1−12.6T+61T2 1 - 12.6T + 61T^{2}
67 1+4.51T+67T2 1 + 4.51T + 67T^{2}
71 1+13.5T+71T2 1 + 13.5T + 71T^{2}
73 1+4.10T+73T2 1 + 4.10T + 73T^{2}
79 1+8.20T+79T2 1 + 8.20T + 79T^{2}
83 1+6.36T+83T2 1 + 6.36T + 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1−11.5T+97T2 1 - 11.5T + 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

−11.53824443193448835165892721793, −10.13747278087485160967299857856, −9.234744011646566970448844250378, −9.057157262933332291332071696904, −7.26177513710809638175320118686, −6.26386634874263879136623074131, −5.48811314175446666034803490128, −4.28894803860738802833264060397, −3.24014193388489653431117946010, −1.92556801346990797493138724393, 1.92556801346990797493138724393, 3.24014193388489653431117946010, 4.28894803860738802833264060397, 5.48811314175446666034803490128, 6.26386634874263879136623074131, 7.26177513710809638175320118686, 9.057157262933332291332071696904, 9.234744011646566970448844250378, 10.13747278087485160967299857856, 11.53824443193448835165892721793

Graph of the ZZ-function along the critical line