↑

Properties

Label 2-403-31.4-c1-0-2
Degree $2$
Conductor $403$
Sign $-0.0238 + 0.999i$
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  + (0.700 + 2.15i)2-s + (−0.672 + 2.07i)3-s + (−2.54 + 1.84i)4-s − 3.69·5-s − 4.93·6-s + (0.888 − 0.645i)7-s + (−2.09 − 1.52i)8-s + (−1.40 − 1.02i)9-s + (−2.59 − 7.97i)10-s + (0.853 − 0.619i)11-s + (−2.11 − 6.50i)12-s + (−0.309 + 0.951i)13-s + (2.01 + 1.46i)14-s + (2.48 − 7.65i)15-s + (−0.126 + 0.390i)16-s + (0.552 + 0.401i)17-s + ⋯
L(s)  = 1  + (0.495 + 1.52i)2-s + (−0.388 + 1.19i)3-s + (−1.27 + 0.923i)4-s − 1.65·5-s − 2.01·6-s + (0.335 − 0.243i)7-s + (−0.740 − 0.538i)8-s + (−0.468 − 0.340i)9-s + (−0.819 − 2.52i)10-s + (0.257 − 0.186i)11-s + (−0.610 − 1.87i)12-s + (−0.0857 + 0.263i)13-s + (0.538 + 0.391i)14-s + (0.642 − 1.97i)15-s + (−0.0317 + 0.0976i)16-s + (0.134 + 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0238 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0238 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0238 + 0.999i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.0238 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558168 - 0.571636i\)
\(L(\frac12)\) \(\approx\) \(0.558168 - 0.571636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (5.56 - 0.0773i)T \)
good2 \( 1 + (-0.700 - 2.15i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.672 - 2.07i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
7 \( 1 + (-0.888 + 0.645i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.853 + 0.619i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-0.552 - 0.401i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.56 - 4.82i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (6.03 + 4.38i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.805 + 2.47i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 1.72T + 37T^{2} \)
41 \( 1 + (1.60 + 4.94i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.41 - 7.43i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.0982 - 0.302i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.09 - 5.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.691 + 2.12i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 + (-0.475 - 0.345i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (13.5 - 9.81i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.50 - 1.81i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.03 - 3.18i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.91 - 5.02i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.391 + 0.284i)T + (29.9 - 92.2i)T^{2} \)
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

−11.85406709336364753646588717184, −11.09068576027282684805413642059, −10.14387530570730644189751261695, −8.850344625854523021839148717329, −7.954054881445530123152590038498, −7.40254757604746189011218485099, −6.15026219053357202209983617003, −5.12623405301601987784221314432, −4.07743645809652226206395175594, −3.91622812872993369377851433115, 0.47942501303645790353528607711, 1.88265871896093645802785728355, 3.32411910088437513057032780255, 4.26285084763129231144222828178, 5.42976077247940352279997438265, 7.05690879356853133861981195077, 7.63520485269444892012860035018, 8.774422919465069226151248796963, 10.04253976063438033519551037183, 11.29004257681993457127918656898

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