↑

Properties

Label 2-399-399.368-c0-0-1
Degree $2$
Conductor $399$
Sign $0.962 + 0.272i$
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  + (0.866 + 0.5i)2-s − i·3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s − 7-s − i·8-s − 9-s + (0.499 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (0.5 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)18-s − 19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s − i·3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s − 7-s − i·8-s − 9-s + (0.499 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (0.5 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215233058\)
\(L(\frac12)\) \(\approx\) \(1.215233058\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−11.94133407713978889789906843901, −10.43427383692549433059716172766, −9.656280983374914263968080803170, −8.750360566421614150324186743059, −7.02035466759971569890897436090, −6.64081031191672360058951380505, −6.07323700748384396683247468041, −4.80292245335500407963384309576, −3.37492287204369616497130333167, −1.92570358949887836940424250442, 2.54711281211923991453319588759, 3.56980310083207616281874475535, 4.51173667232136586481213867273, 5.58281937905273413207890281879, 6.25244521971906606452836170233, 8.135247604411387632516389378470, 9.112429467132631612599598184466, 9.727046744871633807475394138199, 10.68303405605277952568508856977, 11.59044705499146941614976761949

Graph of the $Z$-function along the critical line