↑

Properties

Label 2-399-399.23-c0-0-0
Degree $2$
Conductor $399$
Sign $0.520 + 0.853i$
Analytic cond. $0.199126$
Root an. cond. $0.446236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ 0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9869405002\)
\(L(\frac12)\) \(\approx\) \(0.9869405002\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 - T \)
good2 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 - 1.53T + T^{2} \)
37 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line