↑

Properties

Label 2-399-399.23-c0-0-0
Degree 22
Conductor 399399
Sign 0.520+0.853i0.520 + 0.853i
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

Related objects

Downloads

Learn more

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 + (−0.173 + 0.300i)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 + (−0.173 + 0.300i)37-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((0.520+0.853i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s) L(s)=((0.520+0.853i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: 0.520+0.853i0.520 + 0.853i
Analytic conductor: 0.1991260.199126
Root analytic conductor: 0.4462360.446236
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ399(23,⋅)\chi_{399} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :0), 0.520+0.853i)(2,\ 399,\ (\ :0),\ 0.520 + 0.853i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.98694050020.9869405002
L(12)L(\frac12) ≈\approx 0.98694050020.9869405002
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+(0.173−0.300i)T+(−0.5−0.866i)T2 1 + (0.173 - 0.300i)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+(1.43−0.524i)T+(0.766−0.642i)T2 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}
67 1+(1.43+1.20i)T+(0.173+0.984i)T2 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2}
71 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
73 1+(1.43−1.20i)T+(0.173−0.984i)T2 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2}
79 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)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}
show more
show less
   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

−11.65388835762136306718911184171, −10.12987523294358463741241551597, −9.458016324814222809280850053292, −8.886051982436442589450755766709, −7.43006042782346275320645098811, −6.64534322423708791502132155287, −5.89547803296341407635996151920, −4.37169294928854792689804819873, −2.89481171159268113789158899770, −1.67771027132164204761216497882, 2.79526511166598956516704248833, 3.34196033006767641010958111520, 4.50483999946554104953652627800, 5.90412972929902471020237332142, 7.37758711517238289398116608597, 7.890344405000500757148095969704, 8.876756224890577258972414370848, 9.908526811813382892051806690837, 10.44200408481461468316023274337, 11.70913578814595025935479561324

Graph of the ZZ-function along the critical line