↑

Properties

Label 2-1568-224.5-c0-0-0
Degree $2$
Conductor $1568$
Sign $-0.260 + 0.965i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (−1.83 + 0.241i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 − 0.707i)22-s + (−0.366 − 1.36i)23-s + (0.258 − 0.965i)25-s + (−0.707 − 1.70i)29-s + (−0.258 + 0.965i)32-s + (−0.707 − 0.707i)36-s + (−0.607 + 0.465i)37-s + (−0.292 + 0.707i)43-s + (−1.46 + 1.12i)44-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (−1.83 + 0.241i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 − 0.707i)22-s + (−0.366 − 1.36i)23-s + (0.258 − 0.965i)25-s + (−0.707 − 1.70i)29-s + (−0.258 + 0.965i)32-s + (−0.707 − 0.707i)36-s + (−0.607 + 0.465i)37-s + (−0.292 + 0.707i)43-s + (−1.46 + 1.12i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.260 + 0.965i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ -0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3892512431\)
\(L(\frac12)\) \(\approx\) \(0.3892512431\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (0.258 + 0.965i)T^{2} \)
5 \( 1 + (-0.258 + 0.965i)T^{2} \)
11 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.607 - 0.465i)T + (0.258 - 0.965i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.965 - 0.258i)T^{2} \)
67 \( 1 + (-0.465 + 0.607i)T + (-0.258 - 0.965i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
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

−9.436163299075596833179295757012, −8.399984123239531704176454419334, −8.059654085718082738826007139145, −7.08436151531452506555273162212, −6.26800547724263627940775958212, −5.56098766566245893634039703987, −4.45943465541521444373428612348, −2.98273937392826191919327881469, −2.19954351432691813571606637911, −0.38957572206392821768195466533, 1.70656024992243671519218408500, 2.69519748392960791656496228892, 3.57955079712986227932811022244, 5.22438611303874752698985188220, 5.60805921312780955326767939693, 7.11751592206875705879297595853, 7.54591692616033643294732093841, 8.305683767822369227296962195623, 8.990902434580120092674167658793, 9.942733567449073815387664335752

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