↑

Properties

Label 2-712-712.107-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.0176 - 0.999i$
Analytic cond. $0.355334$
Root an. cond. $0.596099$
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  + (0.654 + 0.755i)2-s + (0.559 + 0.418i)3-s + (−0.142 + 0.989i)4-s + (0.0498 + 0.697i)6-s + (−0.841 + 0.540i)8-s + (−0.144 − 0.490i)9-s + (0.909 + 0.584i)11-s + (−0.494 + 0.494i)12-s + (−0.959 − 0.281i)16-s + (−0.989 − 0.857i)17-s + (0.276 − 0.430i)18-s + (−0.373 − 0.203i)19-s + (0.153 + 1.07i)22-s + (−0.697 − 0.0498i)24-s + (−0.415 + 0.909i)25-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (0.559 + 0.418i)3-s + (−0.142 + 0.989i)4-s + (0.0498 + 0.697i)6-s + (−0.841 + 0.540i)8-s + (−0.144 − 0.490i)9-s + (0.909 + 0.584i)11-s + (−0.494 + 0.494i)12-s + (−0.959 − 0.281i)16-s + (−0.989 − 0.857i)17-s + (0.276 − 0.430i)18-s + (−0.373 − 0.203i)19-s + (0.153 + 1.07i)22-s + (−0.697 − 0.0498i)24-s + (−0.415 + 0.909i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0176 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.0176 - 0.999i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.0176 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478354529\)
\(L(\frac12)\) \(\approx\) \(1.478354529\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (-0.559 - 0.418i)T + (0.281 + 0.959i)T^{2} \)
5 \( 1 + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.909 + 0.415i)T^{2} \)
11 \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.281 + 0.959i)T^{2} \)
17 \( 1 + (0.989 + 0.857i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.373 + 0.203i)T + (0.540 + 0.841i)T^{2} \)
23 \( 1 + (0.540 + 0.841i)T^{2} \)
29 \( 1 + (-0.909 - 0.415i)T^{2} \)
31 \( 1 + (0.540 - 0.841i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.847 + 1.13i)T + (-0.281 + 0.959i)T^{2} \)
43 \( 1 + (-1.94 + 0.424i)T + (0.909 - 0.415i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.114 - 0.0855i)T + (0.281 - 0.959i)T^{2} \)
61 \( 1 + (-0.755 - 0.654i)T^{2} \)
67 \( 1 + (-0.0801 - 0.557i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (-1.74 - 0.512i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.114 - 1.59i)T + (-0.989 + 0.142i)T^{2} \)
97 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
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

−11.00233496114232303031111811842, −9.526933714519028639902868445474, −9.146505201597515644450778308280, −8.288287112904915972025449126071, −7.13210131258764936278332439312, −6.57858593326409785509955167941, −5.42180083046326640442121700770, −4.33309698698171464359716986826, −3.67033557484111286334084862494, −2.44044882206428685175121067365, 1.61745954927173754872982462118, 2.64240277343721040094180467435, 3.79522274678459577490522500100, 4.69112155324992936624992568172, 5.99297532541099505225463794933, 6.63055941871353361677202439105, 8.009936558423526325644290904123, 8.760277816752352989403393939973, 9.603582127285937841876291084370, 10.71359201733046520096223498425

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