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Properties

Label 2-3200-5.4-c1-0-66
Degree $2$
Conductor $3200$
Sign $-0.894 + 0.447i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
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  + 2i·3-s − 4i·7-s − 9-s − 2·11-s − 2i·13-s + 2i·17-s − 2·19-s + 8·21-s − 4i·23-s + 4i·27-s − 6·29-s − 4i·33-s + 10i·37-s + 4·39-s − 6·41-s + ⋯
L(s)  = 1  + 1.15i·3-s − 1.51i·7-s − 0.333·9-s − 0.603·11-s − 0.554i·13-s + 0.485i·17-s − 0.458·19-s + 1.74·21-s − 0.834i·23-s + 0.769i·27-s − 1.11·29-s − 0.696i·33-s + 1.64i·37-s + 0.640·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -0.894 + 0.447i)\)

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)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 14T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.334441302051277550921406765719, −7.67455813663108951542839391754, −6.90543675263443085225841505650, −6.02551992111356855047994170118, −4.97321831644272694877932614841, −4.45771319679722568751419695004, −3.72681319535339418447199570017, −2.97935431906957312171637388271, −1.46038698318094762408882154618, 0, 1.68107224980980954555286641825, 2.23103731694955860958888037925, 3.16317372166788583079257348997, 4.42839328559680894651992689943, 5.50003191517293580533629521356, 5.90041518263471523355653791751, 6.82976290048234225091642455929, 7.48291971878980060725313575119, 8.114914068764887856369628062794

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