↑

Properties

Label 2-1288-1288.1021-c0-0-1
Degree $2$
Conductor $1288$
Sign $0.986 + 0.165i$
Analytic cond. $0.642795$
Root an. cond. $0.801745$
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  + (0.959 − 0.281i)2-s + (−0.989 − 1.14i)3-s + (0.841 − 0.540i)4-s + (0.258 + 1.80i)5-s + (−1.27 − 0.817i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.182 + 1.27i)9-s + (0.755 + 1.65i)10-s + (−1.45 − 0.425i)12-s + (0.234 + 0.512i)13-s + (−0.142 + 0.989i)14-s + (1.80 − 2.07i)15-s + (0.415 − 0.909i)16-s + (0.182 + 1.27i)18-s + (1.66 − 1.07i)19-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (−0.989 − 1.14i)3-s + (0.841 − 0.540i)4-s + (0.258 + 1.80i)5-s + (−1.27 − 0.817i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.182 + 1.27i)9-s + (0.755 + 1.65i)10-s + (−1.45 − 0.425i)12-s + (0.234 + 0.512i)13-s + (−0.142 + 0.989i)14-s + (1.80 − 2.07i)15-s + (0.415 − 0.909i)16-s + (0.182 + 1.27i)18-s + (1.66 − 1.07i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451531988\)
\(L(\frac12)\) \(\approx\) \(1.451531988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (0.989 + 1.14i)T + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.234 - 0.512i)T + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.142 - 0.989i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.822 + 1.80i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.153 - 1.07i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.959 - 0.281i)T^{2} \)
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

−10.10764670692028205002815721235, −9.325599669211276031538777644987, −7.51891877776549511685410196067, −7.08645453849821531629958740927, −6.40302897614407250609496163527, −5.86849077540390296423366897362, −5.10460425918801629432737092190, −3.40220549385777962417881113914, −2.72420880170324863838580396742, −1.68424333045715390627349916225, 1.18314021850846816130818321708, 3.32388968276794197215932281500, 4.26329281015070613495781713249, 4.73490130700908607826412946743, 5.58744779201322369697138921506, 5.97680918984370739304459769223, 7.31178019215491637802098002684, 8.214058467378178903663599001228, 9.226740457499621734108106362186, 10.05920238697807128866610658062

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