↑

Properties

Label 2-403-403.61-c0-0-0
Degree $2$
Conductor $403$
Sign $-0.252 - 0.967i$
Analytic cond. $0.201123$
Root an. cond. $0.448467$
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  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 − 0.499i)6-s + (0.5 + 0.866i)7-s − 8-s + (0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + (−0.866 + 0.499i)22-s + (0.866 + 0.5i)23-s + (0.866 + 0.499i)24-s − 25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 − 0.499i)6-s + (0.5 + 0.866i)7-s − 8-s + (0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s − 0.999i·21-s + (−0.866 + 0.499i)22-s + (0.866 + 0.5i)23-s + (0.866 + 0.499i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(0.201123\)
Root analytic conductor: \(0.448467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :0),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5110509072\)
\(L(\frac12)\) \(\approx\) \(0.5110509072\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - iT \)
31 \( 1 + iT \)
good2 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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

−11.77530506557138552346211898432, −11.23139053739862657298438999019, −9.478265151082825316496414339323, −9.005738537592394110072855525459, −7.88055998241061986020108531661, −6.99341365948518465897903539028, −6.19859338694373332161603244505, −5.53310492123583210019487167153, −3.94653302738206801344771665441, −1.97586661485529291718951358492, 0.977486845819645189753550122434, 2.82270671716034486608668257237, 4.26209016594983671575054917896, 5.35443994116059322496251342488, 6.34264288123609241439954148522, 7.54937358194585242573163224180, 8.854098881384872296013163191604, 9.641464993681278282503144634386, 10.69469974176932338790855305791, 11.05980285643384473421313526746

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