↑

Properties

Label 2-403-13.4-c1-0-9
Degree $2$
Conductor $403$
Sign $0.999 - 0.0400i$
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  + (−2.14 − 1.24i)2-s + (−0.331 + 0.574i)3-s + (2.08 + 3.60i)4-s + 0.685i·5-s + (1.42 − 0.823i)6-s + (3.86 − 2.23i)7-s − 5.36i·8-s + (1.27 + 2.21i)9-s + (0.850 − 1.47i)10-s + (−4.37 − 2.52i)11-s − 2.76·12-s + (−1.71 + 3.17i)13-s − 11.0·14-s + (−0.393 − 0.227i)15-s + (−2.49 + 4.32i)16-s + (3.04 + 5.27i)17-s + ⋯
L(s)  = 1  + (−1.51 − 0.877i)2-s + (−0.191 + 0.331i)3-s + (1.04 + 1.80i)4-s + 0.306i·5-s + (0.582 − 0.336i)6-s + (1.46 − 0.844i)7-s − 1.89i·8-s + (0.426 + 0.738i)9-s + (0.268 − 0.465i)10-s + (−1.32 − 0.762i)11-s − 0.797·12-s + (−0.476 + 0.879i)13-s − 2.96·14-s + (−0.101 − 0.0587i)15-s + (−0.623 + 1.08i)16-s + (0.738 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.999 - 0.0400i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.999 - 0.0400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681325 + 0.0136563i\)
\(L(\frac12)\) \(\approx\) \(0.681325 + 0.0136563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.71 - 3.17i)T \)
31 \( 1 - iT \)
good2 \( 1 + (2.14 + 1.24i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.331 - 0.574i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.685iT - 5T^{2} \)
7 \( 1 + (-3.86 + 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.37 + 2.52i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.93 + 1.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.10 - 1.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.54 + 2.67i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (0.847 + 0.489i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.98 - 5.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.440 + 0.763i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.67iT - 47T^{2} \)
53 \( 1 - 7.71T + 53T^{2} \)
59 \( 1 + (-0.266 + 0.154i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.17 - 7.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.20 - 2.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.8 + 7.98i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 + 3.06T + 79T^{2} \)
83 \( 1 + 1.29iT - 83T^{2} \)
89 \( 1 + (15.8 + 9.13i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.07 + 3.50i)T + (48.5 - 84.0i)T^{2} \)
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

−10.96610993468485627267406996099, −10.47951212714009945625384702724, −9.785792040420515445402157324012, −8.423317294469889707761330849533, −7.891287035177212784064134338588, −7.21665506990037639784817344217, −5.32545267171577839845404084619, −4.11611153131114777737076433476, −2.54018345792545753889942175978, −1.30594122627089275356306880061, 0.884945103928945286263023322812, 2.35440556800490507676851454021, 5.05804541291604409004933991585, 5.53440738096415269936236634174, 7.02257589158476934467131242145, 7.69552967282267763980423716640, 8.299043619894173722660493362305, 9.301459740706594260976828402382, 10.06533776995145542996046292299, 10.95516363703472832418513166471

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