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Properties

Label 2-399-1.1-c1-0-10
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  + 2.09·2-s − 3-s + 2.36·4-s + 0.388·5-s − 2.09·6-s + 7-s + 0.771·8-s + 9-s + 0.811·10-s + 6.41·11-s − 2.36·12-s + 3.88·13-s + 2.09·14-s − 0.388·15-s − 3.12·16-s − 4.98·17-s + 2.09·18-s − 19-s + 0.919·20-s − 21-s + 13.4·22-s + 3.44·23-s − 0.771·24-s − 4.84·25-s + 8.12·26-s − 27-s + 2.36·28-s + ⋯
L(s)  = 1  + 1.47·2-s − 0.577·3-s + 1.18·4-s + 0.173·5-s − 0.853·6-s + 0.377·7-s + 0.272·8-s + 0.333·9-s + 0.256·10-s + 1.93·11-s − 0.683·12-s + 1.07·13-s + 0.558·14-s − 0.100·15-s − 0.781·16-s − 1.20·17-s + 0.492·18-s − 0.229·19-s + 0.205·20-s − 0.218·21-s + 2.86·22-s + 0.717·23-s − 0.157·24-s − 0.969·25-s + 1.59·26-s − 0.192·27-s + 0.447·28-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 2.6829868772.682986877
L(12)L(\frac12) ≈\approx 2.6829868772.682986877
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 1−2.09T+2T2 1 - 2.09T + 2T^{2}
5 1−0.388T+5T2 1 - 0.388T + 5T^{2}
11 1−6.41T+11T2 1 - 6.41T + 11T^{2}
13 1−3.88T+13T2 1 - 3.88T + 13T^{2}
17 1+4.98T+17T2 1 + 4.98T + 17T^{2}
23 1−3.44T+23T2 1 - 3.44T + 23T^{2}
29 1−0.169T+29T2 1 - 0.169T + 29T^{2}
31 1+8.62T+31T2 1 + 8.62T + 31T^{2}
37 1−7.37T+37T2 1 - 7.37T + 37T^{2}
41 1+8.77T+41T2 1 + 8.77T + 41T^{2}
43 1+9.11T+43T2 1 + 9.11T + 43T^{2}
47 1+4.80T+47T2 1 + 4.80T + 47T^{2}
53 1−8.42T+53T2 1 - 8.42T + 53T^{2}
59 1−2.97T+59T2 1 - 2.97T + 59T^{2}
61 1−5.82T+61T2 1 - 5.82T + 61T^{2}
67 1+14.9T+67T2 1 + 14.9T + 67T^{2}
71 1−4.24T+71T2 1 - 4.24T + 71T^{2}
73 1−13.5T+73T2 1 - 13.5T + 73T^{2}
79 1+1.01T+79T2 1 + 1.01T + 79T^{2}
83 1+4.32T+83T2 1 + 4.32T + 83T^{2}
89 1−13.8T+89T2 1 - 13.8T + 89T^{2}
97 1+13.7T+97T2 1 + 13.7T + 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

−11.47685096821963154885527199474, −10.97232803514642848210651427701, −9.443386790050601825531005598223, −8.609016414617399720980204643918, −6.86071787260097522949000571741, −6.37544928927090577062163417151, −5.41264417966402576008142593713, −4.30434887053437066504023806442, −3.63340731583608311598654472987, −1.75514685784801947870251495642, 1.75514685784801947870251495642, 3.63340731583608311598654472987, 4.30434887053437066504023806442, 5.41264417966402576008142593713, 6.37544928927090577062163417151, 6.86071787260097522949000571741, 8.609016414617399720980204643918, 9.443386790050601825531005598223, 10.97232803514642848210651427701, 11.47685096821963154885527199474

Graph of the ZZ-function along the critical line