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Properties

Label 2-399-399.110-c0-0-0
Degree 22
Conductor 399399
Sign −0.585−0.810i-0.585 - 0.810i
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  + (−0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.326 + 1.85i)13-s + (−0.939 − 0.342i)16-s − 19-s + (0.5 − 0.866i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.984i)28-s + 1.53·31-s + (0.939 + 0.342i)36-s + (1.70 + 0.984i)37-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)12-s + (−0.326 + 1.85i)13-s + (−0.939 − 0.342i)16-s − 19-s + (0.5 − 0.866i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.984i)28-s + 1.53·31-s + (0.939 + 0.342i)36-s + (1.70 + 0.984i)37-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((−0.585−0.810i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s) L(s)=((−0.585−0.810i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: −0.585−0.810i-0.585 - 0.810i
Analytic conductor: 0.1991260.199126
Root analytic conductor: 0.4462360.446236
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ399(110,⋅)\chi_{399} (110, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :0), −0.585−0.810i)(2,\ 399,\ (\ :0),\ -0.585 - 0.810i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50982472570.5098247257
L(12)L(\frac12) ≈\approx 0.50982472570.5098247257
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+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
7 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
19 1+T 1 + T
good2 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
5 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}
17 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
23 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
29 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
31 1−1.53T+T2 1 - 1.53T + T^{2}
37 1+(−1.70−0.984i)T+(0.5+0.866i)T2 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2}
41 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
43 1+(0.266−0.223i)T+(0.173−0.984i)T2 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2}
47 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
53 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
59 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
61 1+(0.439+1.20i)T+(−0.766+0.642i)T2 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2}
67 1+(−0.439+0.524i)T+(−0.173−0.984i)T2 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2}
71 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
73 1+(−0.439−0.524i)T+(−0.173+0.984i)T2 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2}
79 1+(0.439−1.20i)T+(−0.766−0.642i)T2 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2}
83 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
89 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
97 1+(0.939+0.342i)T+(0.766+0.642i)T2 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)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

−11.86659116262260369093287250517, −11.08132644281398497784701044627, −9.841972604950179721925688965407, −9.241813082284408764596729718613, −8.325782184886548084124497698111, −6.75857442434159908319389917089, −6.40642193289514643874027728686, −4.73715340245241738788994312315, −4.05368792781209756199380944555, −2.73316157442354504973908410888, 0.78169091650337380349031349174, 2.68684156072379915195060162685, 4.52557202555545190941733793829, 5.62912528322123446992134405007, 6.27230168895615760336101874052, 7.21763010389751902209129431214, 8.335241189732926326185990298982, 9.663981819894624684256573503254, 10.45387672179324208923031345690, 10.89519750699449590781682960237

Graph of the ZZ-function along the critical line