Properties

Label 2-403-403.16-c1-0-6
Degree $2$
Conductor $403$
Sign $0.727 - 0.685i$
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  + (−0.187 − 0.0832i)2-s + (−0.0175 + 0.167i)3-s + (−1.31 − 1.45i)4-s − 4.19·5-s + (0.0171 − 0.0297i)6-s + (1.71 + 1.90i)7-s + (0.250 + 0.770i)8-s + (2.90 + 0.617i)9-s + (0.784 + 0.349i)10-s + (2.54 − 0.540i)11-s + (0.266 − 0.193i)12-s + (−3.60 − 0.0795i)13-s + (−0.162 − 0.499i)14-s + (0.0736 − 0.700i)15-s + (−0.392 + 3.72i)16-s + (5.68 + 1.20i)17-s + ⋯
L(s)  = 1  + (−0.132 − 0.0588i)2-s + (−0.0101 + 0.0964i)3-s + (−0.655 − 0.727i)4-s − 1.87·5-s + (0.00702 − 0.0121i)6-s + (0.648 + 0.720i)7-s + (0.0885 + 0.272i)8-s + (0.968 + 0.205i)9-s + (0.247 + 0.110i)10-s + (0.767 − 0.163i)11-s + (0.0768 − 0.0558i)12-s + (−0.999 − 0.0220i)13-s + (−0.0433 − 0.133i)14-s + (0.0190 − 0.180i)15-s + (−0.0980 + 0.932i)16-s + (1.37 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767166 + 0.304469i\)
\(L(\frac12)\) \(\approx\) \(0.767166 + 0.304469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.60 + 0.0795i)T \)
31 \( 1 + (-2.11 - 5.14i)T \)
good2 \( 1 + (0.187 + 0.0832i)T + (1.33 + 1.48i)T^{2} \)
3 \( 1 + (0.0175 - 0.167i)T + (-2.93 - 0.623i)T^{2} \)
5 \( 1 + 4.19T + 5T^{2} \)
7 \( 1 + (-1.71 - 1.90i)T + (-0.731 + 6.96i)T^{2} \)
11 \( 1 + (-2.54 + 0.540i)T + (10.0 - 4.47i)T^{2} \)
17 \( 1 + (-5.68 - 1.20i)T + (15.5 + 6.91i)T^{2} \)
19 \( 1 + (-0.420 - 3.99i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (0.257 - 0.285i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-1.02 - 0.454i)T + (19.4 + 21.5i)T^{2} \)
37 \( 1 + (-2.51 - 4.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.25 + 3.67i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (-1.14 - 10.8i)T + (-42.0 + 8.94i)T^{2} \)
47 \( 1 + (2.84 + 2.06i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.19 + 6.74i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.202 + 0.0903i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-2.73 + 4.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.60 - 6.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.63 + 0.559i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (1.68 - 5.19i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.62 - 4.99i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.9 - 7.95i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-9.42 + 2.00i)T + (81.3 - 36.1i)T^{2} \)
97 \( 1 + (0.562 + 0.624i)T + (-10.1 + 96.4i)T^{2} \)
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

−11.53261284569842570316111565501, −10.38354180177694390496446618933, −9.654288069056975226086418796278, −8.391220441407563044990340802155, −7.968512988716871044398427402277, −6.82195543574285982493842355724, −5.26487736725721853148754370545, −4.52053085178702150082594824733, −3.53419808458481834486489553943, −1.32265547578419567928154497206, 0.70525616476631491103886678187, 3.33999247366945064205528172433, 4.25743531991012849314186660770, 4.73758922791030548072151241759, 7.10874542221257996547177938329, 7.43618996416741334851158412690, 8.115981914776407223762540947066, 9.215935872971930557950629877735, 10.20441640827456472359969284454, 11.44883345218183631655183354234

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