↑

Properties

Label 2-409-409.217-c1-0-23
Degree $2$
Conductor $409$
Sign $-0.809 - 0.586i$
Analytic cond. $3.26588$
Root an. cond. $1.80717$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (−0.633 + 0.366i)3-s + (0.5 + 0.866i)4-s + 1.73·5-s + 1.26·6-s + (−1 − 0.267i)7-s + 1.73i·8-s + (−1.23 + 2.13i)9-s + (−2.59 − 1.49i)10-s + (−3 − 3i)11-s + (−0.633 − 0.366i)12-s + (−0.366 − 0.366i)13-s + (1.26 + 1.26i)14-s + (−1.09 + 0.633i)15-s + (2.49 − 4.33i)16-s + (0.232 − 0.401i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (−0.366 + 0.211i)3-s + (0.250 + 0.433i)4-s + 0.774·5-s + 0.517·6-s + (−0.377 − 0.101i)7-s + 0.612i·8-s + (−0.410 + 0.711i)9-s + (−0.821 − 0.474i)10-s + (−0.904 − 0.904i)11-s + (−0.183 − 0.105i)12-s + (−0.101 − 0.101i)13-s + (0.338 + 0.338i)14-s + (−0.283 + 0.163i)15-s + (0.624 − 1.08i)16-s + (0.0562 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(409\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(3.26588\)
Root analytic conductor: \(1.80717\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{409} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 409,\ (\ :1/2),\ -0.809 - 0.586i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad409 \( 1 + (3 - 20i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.633 - 0.366i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + (1 + 0.267i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
13 \( 1 + (0.366 + 0.366i)T + 13iT^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 + 2.73i)T + 19iT^{2} \)
23 \( 1 + (1.09 + 0.633i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.90 - 7.09i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.73 - 5.73i)T - 31iT^{2} \)
37 \( 1 + (8.09 - 2.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.19 - 4.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 2i)T - 43iT^{2} \)
47 \( 1 + (-3 + 11.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.46 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.73 - 7.73i)T + 59iT^{2} \)
61 \( 1 + (-5.63 + 5.63i)T - 61iT^{2} \)
67 \( 1 + (12.1 + 3.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.43 + 9.09i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5 - 5i)T + 79iT^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 + (-0.794 + 2.96i)T + (-84.0 - 48.5i)T^{2} \)
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

−10.52515209030335247878444626230, −10.08411765669315801576183141838, −8.889594156298823674529401527882, −8.390706831822321560923472979260, −7.09324414930802075461270773300, −5.66683179681231655092417310126, −5.13759453904436590228077958397, −3.09059570446152166599001616557, −1.93142442332244212890297891194, 0, 2.03565977249798668737482206862, 3.82887526263359154591352525412, 5.54452019232569610415928727519, 6.27323663351687743758373230250, 7.20098318228978586570759475418, 8.088965167745127683422913620832, 9.143282685758708723354895120248, 9.802416914325194478142978889710, 10.42975080398283563038752709603

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