↑

Properties

Label 2-1564-1564.1223-c0-0-1
Degree $2$
Conductor $1564$
Sign $-0.952 - 0.305i$
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  + (0.415 + 0.909i)2-s + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.186 + 0.215i)6-s + (0.186 + 1.29i)7-s + (−0.959 − 0.281i)8-s + (−0.773 + 0.496i)9-s + (−0.797 + 1.74i)11-s + (−0.118 + 0.258i)12-s + (0.186 − 1.29i)13-s + (−1.10 + 0.708i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−0.773 − 0.496i)18-s + (0.154 + 0.339i)21-s − 1.91·22-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.186 + 0.215i)6-s + (0.186 + 1.29i)7-s + (−0.959 − 0.281i)8-s + (−0.773 + 0.496i)9-s + (−0.797 + 1.74i)11-s + (−0.118 + 0.258i)12-s + (0.186 − 1.29i)13-s + (−1.10 + 0.708i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−0.773 − 0.496i)18-s + (0.154 + 0.339i)21-s − 1.91·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087304757\)
\(L(\frac12)\) \(\approx\) \(1.087304757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.830 - 1.81i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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

−9.747900285421762962044915203636, −8.897829516309000022327402744948, −8.255282862975311927596193657359, −7.68292575983350579818578258237, −6.79593376406128167915606683837, −5.74137870428161482626032827821, −5.17065668553531068607147749645, −4.50840165735720587862679759654, −2.87576759909559420010341410952, −2.45209876422008420674718989062, 0.71051724044555093964614809119, 2.21016625017147395214686801366, 3.34702221310977595260068398682, 3.94320298958285459985715363231, 4.82929198538778766359186710720, 6.05103647327584917322833506204, 6.50079884127105418492892363978, 8.053782763593601696968171126958, 8.506485343959601457288436794966, 9.369456989991438851675285674406

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