↑

Properties

Label 2-414-23.4-c1-0-3
Degree $2$
Conductor $414$
Sign $-0.223 - 0.974i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.915 + 0.588i)5-s + (0.122 + 0.854i)7-s + (−0.959 − 0.281i)8-s + (−0.154 + 1.07i)10-s + (−0.273 + 0.598i)11-s + (−0.882 + 6.13i)13-s + (−0.726 + 0.466i)14-s + (−0.142 − 0.989i)16-s + (3.72 + 4.30i)17-s + (4.22 − 4.87i)19-s + (−1.04 + 0.306i)20-s − 0.657·22-s + (−3.73 + 3.01i)23-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−0.327 + 0.377i)4-s + (0.409 + 0.263i)5-s + (0.0464 + 0.323i)7-s + (−0.339 − 0.0996i)8-s + (−0.0489 + 0.340i)10-s + (−0.0823 + 0.180i)11-s + (−0.244 + 1.70i)13-s + (−0.194 + 0.124i)14-s + (−0.0355 − 0.247i)16-s + (0.904 + 1.04i)17-s + (0.969 − 1.11i)19-s + (−0.233 + 0.0685i)20-s − 0.140·22-s + (−0.778 + 0.628i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.223 - 0.974i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.223 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985755 + 1.23781i\)
\(L(\frac12)\) \(\approx\) \(0.985755 + 1.23781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
3 \( 1 \)
23 \( 1 + (3.73 - 3.01i)T \)
good5 \( 1 + (-0.915 - 0.588i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.122 - 0.854i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.273 - 0.598i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.882 - 6.13i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-3.72 - 4.30i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-4.22 + 4.87i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-0.0667 - 0.0769i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-1.48 - 0.434i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (6.17 - 3.96i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (5.12 + 3.29i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-5.15 + 1.51i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 7.84T + 47T^{2} \)
53 \( 1 + (0.676 + 4.70i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.808 + 5.62i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (0.215 + 0.0632i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-0.986 - 2.15i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (2.66 + 5.84i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.73 + 3.15i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-2.41 + 16.8i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-14.2 + 9.18i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-4.85 + 1.42i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (8.27 + 5.31i)T + (40.2 + 88.2i)T^{2} \)
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

−11.80581868668221283178339750056, −10.44738442894233765229416444005, −9.533352515233983307124250459400, −8.745319684654022135221553432290, −7.60654399528981306003174518879, −6.73126722100406834496626257296, −5.84746247546660374449030764738, −4.80814579771672273346627288903, −3.61874278737796150125995329267, −2.05135580726245225845637487900, 1.01511577541889429310235523001, 2.72828296896799565811145122686, 3.78513176333712227997044483708, 5.28431220110118164127142446632, 5.73015337118118673207107700782, 7.38509175329547533681955657026, 8.192464120511675673367827253653, 9.492969222167443852358667197565, 10.12206994796063830871414041228, 10.84979364121282295582032486172

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