↑

Properties

Label 2-414-23.6-c1-0-6
Degree $2$
Conductor $414$
Sign $0.817 + 0.575i$
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 + (−0.273 + 0.175i)5-s + (0.346 − 2.40i)7-s + (0.959 − 0.281i)8-s + (−0.0462 − 0.321i)10-s + (−1.41 − 3.09i)11-s + (−0.0383 − 0.266i)13-s + (2.04 + 1.31i)14-s + (−0.142 + 0.989i)16-s + (3.17 − 3.66i)17-s + (−2.63 − 3.04i)19-s + (0.311 + 0.0914i)20-s + 3.40·22-s + (4.20 − 2.31i)23-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (−0.327 − 0.377i)4-s + (−0.122 + 0.0784i)5-s + (0.130 − 0.909i)7-s + (0.339 − 0.0996i)8-s + (−0.0146 − 0.101i)10-s + (−0.425 − 0.932i)11-s + (−0.0106 − 0.0739i)13-s + (0.546 + 0.351i)14-s + (−0.0355 + 0.247i)16-s + (0.770 − 0.888i)17-s + (−0.604 − 0.697i)19-s + (0.0696 + 0.0204i)20-s + 0.724·22-s + (0.875 − 0.482i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965999 - 0.306037i\)
\(L(\frac12)\) \(\approx\) \(0.965999 - 0.306037i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.20 + 2.31i)T \)
good5 \( 1 + (0.273 - 0.175i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.346 + 2.40i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (1.41 + 3.09i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.0383 + 0.266i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-3.17 + 3.66i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (2.63 + 3.04i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-1.31 + 1.51i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-9.12 + 2.67i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (1.69 + 1.09i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-2.19 + 1.40i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (3.77 + 1.10i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 + (0.153 - 1.06i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.40 - 9.80i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (4.17 - 1.22i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-4.26 + 9.34i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (4.74 - 10.3i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.746 - 0.861i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (2.28 + 15.8i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-0.887 - 0.570i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (2.62 + 0.772i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (11.4 - 7.35i)T + (40.2 - 88.2i)T^{2} \)
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

−10.94493059672999844787075117036, −10.23710412232457432965863830675, −9.208512094210200649505180029947, −8.232512072381289111858588151564, −7.44683357786159586861081761236, −6.58208653690229099276270211089, −5.43143962764199074977105956571, −4.39263155008294320720252862170, −2.99582754847970991752004440852, −0.76577126864484413450757757002, 1.71999146953551942794368333543, 2.95627273224154019564567615930, 4.33890737808217875861874067714, 5.40620523290414953051794460013, 6.63452826130933615022975184090, 7.998675921058277678920022539055, 8.510144007475553989341920675545, 9.718208373166180835441626054384, 10.27688113532558214236037009856, 11.34315240536199216224063830547

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