↑

Properties

Label 2-1564-1564.1291-c0-0-1
Degree $2$
Conductor $1564$
Sign $0.994 - 0.105i$
Analytic cond. $0.780537$
Root an. cond. $0.883480$
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  + (0.142 + 0.989i)2-s + (0.449 − 0.983i)3-s + (−0.959 + 0.281i)4-s + (1.03 + 0.304i)6-s + (−0.474 + 0.304i)7-s + (−0.415 − 0.909i)8-s + (−0.110 − 0.127i)9-s + (0.258 − 1.80i)11-s + (−0.153 + 1.07i)12-s + (1.61 + 1.03i)13-s + (−0.368 − 0.425i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + (0.110 − 0.127i)18-s + (0.0867 + 0.603i)21-s + 1.81·22-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (0.449 − 0.983i)3-s + (−0.959 + 0.281i)4-s + (1.03 + 0.304i)6-s + (−0.474 + 0.304i)7-s + (−0.415 − 0.909i)8-s + (−0.110 − 0.127i)9-s + (0.258 − 1.80i)11-s + (−0.153 + 1.07i)12-s + (1.61 + 1.03i)13-s + (−0.368 − 0.425i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + (0.110 − 0.127i)18-s + (0.0867 + 0.603i)21-s + 1.81·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ 0.994 - 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261851426\)
\(L(\frac12)\) \(\approx\) \(1.261851426\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (-0.281 + 0.959i)T \)
good3 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.474 - 0.304i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (-0.258 + 1.80i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (-0.755 - 1.65i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (1.66 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (0.142 + 0.989i)T^{2} \)
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

−8.960129959968333638303731451360, −8.680713818194178040731404400439, −8.226852707064946905227585663339, −7.01413418689917807732524249487, −6.37748921710239338422496636636, −6.10967614741224205081698469984, −4.72687938695194043299034301683, −3.71262103627516496464627467684, −2.74283402454454958879914257875, −1.10561586277043677162959921956, 1.47757757053354096717780731021, 2.78684303005574373109668417928, 3.81741173015340650330442972495, 4.12970100311591750543200497551, 5.15337525903470894022007596106, 6.18623495364147884312945981205, 7.32880484111456381307867287268, 8.407430904011505150926703260399, 9.152487704757588671168986520272, 9.781838316021002097215219715869

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