Properties

Label 2-1575-5.4-c1-0-12
Degree $2$
Conductor $1575$
Sign $-0.894 - 0.447i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s − 0.999·4-s + i·7-s + 1.73i·8-s + 3.46·11-s − 2i·13-s − 1.73·14-s − 5·16-s + 3.46i·17-s + 4·19-s + 5.99i·22-s + 3.46i·23-s + 3.46·26-s − 0.999i·28-s − 4·31-s − 5.19i·32-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + 0.377i·7-s + 0.612i·8-s + 1.04·11-s − 0.554i·13-s − 0.462·14-s − 1.25·16-s + 0.840i·17-s + 0.917·19-s + 1.27i·22-s + 0.722i·23-s + 0.679·26-s − 0.188i·28-s − 0.718·31-s − 0.918i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.794229894\)
\(L(\frac12)\) \(\approx\) \(1.794229894\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 - 1.73iT - 2T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.308398747430502392410527128761, −8.972188705600403212592253799664, −7.82946417755454683550584674009, −7.51872314135418040241770643226, −6.38419679050228507588335022166, −5.94138654032206328766067264515, −5.11198122289407512063562546281, −4.05638782637406176787541441367, −2.89889407407878171745927379103, −1.48617663958032203974701151566, 0.74074020685023651235706219397, 1.82479676309807203416997139089, 2.90045797645120761497918157154, 3.83961525327585722238628927932, 4.50266705756385738970907081378, 5.74333422097786033000517627882, 6.88775256271281314975552327925, 7.26734266355370240333087551810, 8.644435602915046888891017952852, 9.367012253953055179189984798828

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