↑

Properties

Label 2-1575-35.34-c0-0-4
Degree $2$
Conductor $1575$
Sign $-0.447 + 0.894i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·2-s − 1.99·4-s + i·7-s + 1.73i·8-s + 1.73·11-s + 1.73·14-s + 0.999·16-s − 2.99i·22-s − 1.73i·23-s − 1.99i·28-s + 1.73·29-s − i·37-s + i·43-s − 3.46·44-s − 2.99·46-s + ⋯
L(s)  = 1  − 1.73i·2-s − 1.99·4-s + i·7-s + 1.73i·8-s + 1.73·11-s + 1.73·14-s + 0.999·16-s − 2.99i·22-s − 1.73i·23-s − 1.99i·28-s + 1.73·29-s − i·37-s + i·43-s − 3.46·44-s − 2.99·46-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086922416\)
\(L(\frac12)\) \(\approx\) \(1.086922416\)
\(L(1)\) 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 \( 1 + 1.73iT - T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.73iT - T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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

−9.465126653409923069079830324296, −8.851878032526253824722383164546, −8.350452891115956056815092158325, −6.78795012122613374638110902632, −6.03281488533436179710041809643, −4.75600928554192300715625813404, −4.13528184735617966416694367697, −3.07023543736104858646496743987, −2.27468969674174958043112804023, −1.15584479016556472551739976371, 1.27199221548408011298356897223, 3.49349149987782231262294920555, 4.25862691484670735047819726127, 5.05816797665552691746891672661, 6.14226518754999411028753934037, 6.69136932327867014521648873894, 7.31611576489498468410229755391, 8.070607122214422511812398155339, 8.904750051296440618945204189038, 9.532623661342885182297728751903

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