Properties

Label 2-471-157.156-c1-0-23
Degree $2$
Conductor $471$
Sign $-0.546 + 0.837i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.269i·2-s − 3-s + 1.92·4-s − 3.43i·5-s + 0.269i·6-s − 1.97i·7-s − 1.06i·8-s + 9-s − 0.927·10-s − 5.22·11-s − 1.92·12-s − 0.533·13-s − 0.533·14-s + 3.43i·15-s + 3.56·16-s − 2.09·17-s + ⋯
L(s)  = 1  − 0.190i·2-s − 0.577·3-s + 0.963·4-s − 1.53i·5-s + 0.110i·6-s − 0.746i·7-s − 0.374i·8-s + 0.333·9-s − 0.293·10-s − 1.57·11-s − 0.556·12-s − 0.147·13-s − 0.142·14-s + 0.886i·15-s + 0.891·16-s − 0.508·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ -0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566176 - 1.04614i\)
\(L(\frac12)\) \(\approx\) \(0.566176 - 1.04614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
157 \( 1 + (-6.85 + 10.4i)T \)
good2 \( 1 + 0.269iT - 2T^{2} \)
5 \( 1 + 3.43iT - 5T^{2} \)
7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 + 5.22T + 11T^{2} \)
13 \( 1 + 0.533T + 13T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 + 1.00T + 19T^{2} \)
23 \( 1 - 6.53iT - 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 9.35T + 31T^{2} \)
37 \( 1 + 4.75T + 37T^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 + 9.75iT - 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 11.9iT - 53T^{2} \)
59 \( 1 - 8.86iT - 59T^{2} \)
61 \( 1 - 15.3iT - 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 - 5.06T + 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + 1.66iT - 79T^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 1.25iT - 97T^{2} \)
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

−10.54933001597025768234389914744, −10.23590950963810721392200495216, −8.912410467392071961798798203822, −7.84199662184792545855650416375, −7.15808028723649705926827447816, −5.81015843545247684325267907430, −5.08993420243733088765008309121, −3.96000278989574486680390147895, −2.22201254562933137128204849208, −0.74054186899590103517039732995, 2.40209850609199749176398085518, 2.92660717743251473084291199349, 4.87073048251130179155930990649, 6.04715254848646559836181071138, 6.56955776453147892902511154179, 7.47251015129704860618325192689, 8.340368913083996937702232064177, 9.966071733453935603943101161153, 10.72564767003850070397673363370, 11.01738746412959380979227543918

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