↑

Properties

Label 2-1564-1564.1291-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9139732673\)
\(L(\frac12)\) \(\approx\) \(0.9139732673\)
\(L(1)\) 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

−9.681438273800762845590911139544, −9.441919269180673483420851403699, −8.380061940050689661121015314301, −7.54105606170735192686114547701, −6.80509154920857033278653378198, −5.92468673080924596955677204480, −5.00567422516814376195407313838, −4.21668349775703686089923095734, −3.99634996695436506228547079257, −1.92130090842809939313893319472, 0.78757257141016385497689176826, 1.76158311369799761846198673027, 3.09029979995766359027079529528, 3.82502213310086017982250102024, 5.23692234994416052834669995883, 5.86291802867704184225898914119, 6.53032717982026739722228844521, 7.893764595650719499657709490879, 8.576267652827193586269019311343, 8.961181071718193777426819751860

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