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Properties

Label 2-409-409.360-c1-0-26
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

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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: $\chi_{409} (360, \cdot )$
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 \( 1 + (-0.794 - 2.96i)T + (-84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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