↑

Properties

Label 2-403-13.9-c1-0-6
Degree $2$
Conductor $403$
Sign $-0.542 - 0.840i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.137 − 0.238i)2-s + (−0.732 + 1.26i)3-s + (0.961 + 1.66i)4-s − 1.37·5-s + (0.202 + 0.350i)6-s + (0.239 + 0.415i)7-s + 1.08·8-s + (0.427 + 0.740i)9-s + (−0.189 + 0.328i)10-s + (0.323 − 0.561i)11-s − 2.81·12-s + (−3.42 + 1.12i)13-s + 0.132·14-s + (1.00 − 1.74i)15-s + (−1.77 + 3.07i)16-s + (1.17 + 2.03i)17-s + ⋯
L(s)  = 1  + (0.0975 − 0.168i)2-s + (−0.422 + 0.732i)3-s + (0.480 + 0.833i)4-s − 0.614·5-s + (0.0824 + 0.142i)6-s + (0.0906 + 0.156i)7-s + 0.382·8-s + (0.142 + 0.246i)9-s + (−0.0599 + 0.103i)10-s + (0.0976 − 0.169i)11-s − 0.813·12-s + (−0.949 + 0.312i)13-s + 0.0353·14-s + (0.259 − 0.450i)15-s + (−0.443 + 0.768i)16-s + (0.284 + 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.542 - 0.840i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522868 + 0.960001i\)
\(L(\frac12)\) \(\approx\) \(0.522868 + 0.960001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.42 - 1.12i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.137 + 0.238i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.732 - 1.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.37T + 5T^{2} \)
7 \( 1 + (-0.239 - 0.415i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.323 + 0.561i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.704 + 1.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.994 - 1.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.107 - 0.185i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.54 - 6.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.27 + 9.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.99T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + (-0.838 - 1.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.454 - 0.786i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.20 - 3.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.48 - 4.29i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.35T + 73T^{2} \)
79 \( 1 + 2.73T + 79T^{2} \)
83 \( 1 - 6.40T + 83T^{2} \)
89 \( 1 + (-2.40 + 4.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.40 + 9.36i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.64001491650545729359678870906, −10.78498587701972989274266808739, −9.988248300155668020590053289446, −8.795494841444827130518561825073, −7.77514552363518601753402666611, −7.09819474431407397066202060696, −5.68396386422635660678201939916, −4.47396333466932995801650985188, −3.74178984838032847100235760742, −2.29683951637032539130883731756, 0.71976912544673035618309089835, 2.27981251122676967643305612485, 4.08512994122774697501996812005, 5.34135346321677664066729295474, 6.24630069074887140790680758406, 7.25126177662127938114765019015, 7.67693904613013948911936213325, 9.258746573100143175956905082852, 10.16706889218614735399358290782, 11.04212703683183679238698231568

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