↑

Properties

Label 2-416-104.77-c1-0-3
Degree $2$
Conductor $416$
Sign $-0.376 - 0.926i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + 2.94i·3-s + 1.81·5-s + 1.13i·7-s − 5.70·9-s + 4.40·11-s + (2.58 + 2.50i)13-s + 5.35i·15-s + 0.701·17-s − 5.95·19-s − 3.36·21-s − 4·23-s − 1.70·25-s − 7.96i·27-s − 5.01i·29-s − 8.77i·31-s + ⋯
L(s)  = 1  + 1.70i·3-s + 0.812·5-s + 0.430i·7-s − 1.90·9-s + 1.32·11-s + (0.717 + 0.696i)13-s + 1.38i·15-s + 0.170·17-s − 1.36·19-s − 0.733·21-s − 0.834·23-s − 0.340·25-s − 1.53i·27-s − 0.932i·29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877223 + 1.30374i\)
\(L(\frac12)\) \(\approx\) \(0.877223 + 1.30374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-2.58 - 2.50i)T \)
good3 \( 1 - 2.94iT - 3T^{2} \)
5 \( 1 - 1.81T + 5T^{2} \)
7 \( 1 - 1.13iT - 7T^{2} \)
11 \( 1 - 4.40T + 11T^{2} \)
17 \( 1 - 0.701T + 17T^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 5.01iT - 29T^{2} \)
31 \( 1 + 8.77iT - 31T^{2} \)
37 \( 1 - 3.36T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 2.94iT - 43T^{2} \)
47 \( 1 - 1.13iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 7.49T + 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 8.43iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 2.85T + 83T^{2} \)
89 \( 1 - 8.43iT - 89T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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

−11.30440744252090757877788264870, −10.47108617181400382443526007482, −9.460293674700475995944201224394, −9.274008781034238691658494739359, −8.212496167222911559217333938021, −6.27727532406866439637121668678, −5.86466995308594683462059605868, −4.38908099224828157733290066558, −3.86551578314548073703243082864, −2.19852557076677405497338177757, 1.13490062305638422758992887668, 2.13693133554238527136947475099, 3.73138535482871458166942298399, 5.55641157577867710278831791505, 6.45060146669081730560089938309, 6.92755059805471459677741326407, 8.168941410275845891902757510694, 8.811723498442286920620439813052, 10.08592861607265195604342256998, 11.07345675325993697435141239658

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