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Properties

Label 2-399-399.263-c0-0-0
Degree 22
Conductor 399399
Sign −0.0482+0.998i-0.0482 + 0.998i
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  + (0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)12-s + (−1.43 − 0.524i)13-s + (0.766 + 0.642i)16-s + 19-s + (−0.5 − 0.866i)21-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.939 + 0.342i)28-s + 0.347·31-s + (0.766 + 0.642i)36-s + (0.939 + 1.62i)37-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)12-s + (−1.43 − 0.524i)13-s + (0.766 + 0.642i)16-s + 19-s + (−0.5 − 0.866i)21-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.939 + 0.342i)28-s + 0.347·31-s + (0.766 + 0.642i)36-s + (0.939 + 1.62i)37-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((−0.0482+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s) L(s)=((−0.0482+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: −0.0482+0.998i-0.0482 + 0.998i
Analytic conductor: 0.1991260.199126
Root analytic conductor: 0.4462360.446236
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ399(263,⋅)\chi_{399} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :0), −0.0482+0.998i)(2,\ 399,\ (\ :0),\ -0.0482 + 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.72568755780.7256875578
L(12)L(\frac12) ≈\approx 0.72568755780.7256875578
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+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
7 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
19 1−T 1 - T
good2 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
5 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
17 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
23 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1−0.347T+T2 1 - 0.347T + T^{2}
37 1+(−0.939−1.62i)T+(−0.5+0.866i)T2 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2}
41 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
43 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}
47 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
53 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(−0.266+0.223i)T+(0.173−0.984i)T2 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2}
67 1+(−0.266−1.50i)T+(−0.939+0.342i)T2 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+(−0.266+1.50i)T+(−0.939−0.342i)T2 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2}
79 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
97 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)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

−11.39971121350964962093197447840, −10.22767529525538512009781714566, −9.464273766673610219497398723590, −8.236744825370847393035994705925, −7.75615800283399070077671007313, −6.65714965002717709821980833927, −5.34682232659847370524693912663, −4.54587117556312544680734145058, −2.89670239068255967765801712715, −1.14321656026998932375487104648, 2.58586294249349647645969974647, 3.92077648173447361440386377295, 4.94913824892147118531454796889, 5.44230536664703128485029386056, 7.34591780398307979451048884502, 8.271491775377567494938853959393, 9.202772814669257768945021063750, 9.602873384658710745451509750324, 10.74691189946811093514962714645, 11.77109955343334553069363208068

Graph of the ZZ-function along the critical line