↑

Properties

Label 2-403-403.4-c1-0-4
Degree $2$
Conductor $403$
Sign $-0.710 - 0.704i$
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  + (−0.230 + 1.08i)2-s + (0.418 + 0.465i)3-s + (0.708 + 0.315i)4-s − 1.86i·5-s + (−0.599 + 0.346i)6-s + (−2.06 + 4.63i)7-s + (−1.80 + 2.48i)8-s + (0.272 − 2.59i)9-s + (2.02 + 0.430i)10-s + (−3.75 + 0.394i)11-s + (0.149 + 0.461i)12-s + (1.50 + 3.27i)13-s + (−4.54 − 3.30i)14-s + (0.869 − 0.783i)15-s + (−1.23 − 1.37i)16-s + (−0.577 + 5.49i)17-s + ⋯
L(s)  = 1  + (−0.162 + 0.765i)2-s + (0.241 + 0.268i)3-s + (0.354 + 0.157i)4-s − 0.836i·5-s + (−0.244 + 0.141i)6-s + (−0.780 + 1.75i)7-s + (−0.638 + 0.878i)8-s + (0.0908 − 0.864i)9-s + (0.640 + 0.136i)10-s + (−1.13 + 0.119i)11-s + (0.0432 + 0.133i)12-s + (0.417 + 0.908i)13-s + (−1.21 − 0.882i)14-s + (0.224 − 0.202i)15-s + (−0.309 − 0.343i)16-s + (−0.140 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.710 - 0.704i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.710 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480557 + 1.16726i\)
\(L(\frac12)\) \(\approx\) \(0.480557 + 1.16726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.50 - 3.27i)T \)
31 \( 1 + (-5.00 - 2.44i)T \)
good2 \( 1 + (0.230 - 1.08i)T + (-1.82 - 0.813i)T^{2} \)
3 \( 1 + (-0.418 - 0.465i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + 1.86iT - 5T^{2} \)
7 \( 1 + (2.06 - 4.63i)T + (-4.68 - 5.20i)T^{2} \)
11 \( 1 + (3.75 - 0.394i)T + (10.7 - 2.28i)T^{2} \)
17 \( 1 + (0.577 - 5.49i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-2.66 - 2.40i)T + (1.98 + 18.8i)T^{2} \)
23 \( 1 + (-3.75 + 1.67i)T + (15.3 - 17.0i)T^{2} \)
29 \( 1 + (2.32 + 0.493i)T + (26.4 + 11.7i)T^{2} \)
37 \( 1 + (1.43 + 0.826i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 + 11.8i)T + (-37.4 - 16.6i)T^{2} \)
43 \( 1 + (-3.17 + 3.52i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (9.22 + 2.99i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.98 - 4.34i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.12 - 5.28i)T + (-53.8 + 23.9i)T^{2} \)
61 \( 1 + (3.50 + 6.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.82 + 1.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.38 - 0.986i)T + (69.4 + 14.7i)T^{2} \)
73 \( 1 + (0.663 + 0.913i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-7.35 - 5.34i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-13.0 + 4.24i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.82 + 0.401i)T + (87.0 - 18.5i)T^{2} \)
97 \( 1 + (2.97 - 6.67i)T + (-64.9 - 72.0i)T^{2} \)
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

−11.98614901322222224679156902067, −10.59615024155962185711143462259, −9.280443347427631631210325818885, −8.838181098258326639002125957930, −8.163869814345208907461392409253, −6.74799500518235276797650156235, −5.96887370623298568069363056724, −5.16803291320372258086572904815, −3.47880822068412126031394039821, −2.25982060141920679115706785023, 0.833292604014442797088212054644, 2.79513158286628303547662187685, 3.20368201010284522156011666594, 4.92282403749332905118381143295, 6.47338806113066626964085507167, 7.26373581030309317082248155273, 7.82176515433657698370378179707, 9.613245220053493200939882465780, 10.22996446232286071733246457663, 10.91134734687604915456405027508

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