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Properties

Label 2-399-399.293-c0-0-0
Degree 22
Conductor 399399
Sign −0.0977−0.995i-0.0977 - 0.995i
Analytic cond. 0.1991260.199126
Root an. cond. 0.4462360.446236
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s − 0.999·28-s + 31-s + (0.499 − 0.866i)36-s − 1.73i·37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s − 0.999·28-s + 31-s + (0.499 − 0.866i)36-s − 1.73i·37-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((−0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s) L(s)=((−0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: −0.0977−0.995i-0.0977 - 0.995i
Analytic conductor: 0.1991260.199126
Root analytic conductor: 0.4462360.446236
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ399(293,⋅)\chi_{399} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :0), −0.0977−0.995i)(2,\ 399,\ (\ :0),\ -0.0977 - 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73716849530.7371684953
L(12)L(\frac12) ≈\approx 0.73716849530.7371684953
L(1)L(1) 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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1−T 1 - T
good2 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1−T2 1 - T^{2}
13 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1−T+T2 1 - T + T^{2}
37 1+1.73iT−T2 1 + 1.73iT - T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
67 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
71 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
73 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
79 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{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

−11.72465722849884184555307424374, −10.96019233253088897473532960778, −9.904053640622015743462921526270, −9.127534694388174600124712901249, −8.126019259460014878812871146585, −7.00488337267592848707182301154, −5.92022987538784082001969568899, −5.01560407538597719289788033501, −3.57272492612994412258823854923, −2.74540623192990583307490111422, 1.19629506160355462175283295401, 2.69094705467349542082355555945, 4.57028874424349059741946218838, 5.66331650678136141965320339859, 6.80595339897344670954338681743, 7.01202312709295540649604253045, 8.342560389871093536005179049531, 9.804071045336939630591414474186, 10.30880386432574066113567022219, 11.44220988600568209298668792885

Graph of the ZZ-function along the critical line