↑

Properties

Label 2-471-471.224-c0-0-0
Degree $2$
Conductor $471$
Sign $0.999 + 0.0195i$
Analytic cond. $0.235059$
Root an. cond. $0.484829$
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  + (0.885 − 0.464i)3-s + (0.885 + 0.464i)4-s + (−1.32 + 0.695i)7-s + (0.568 − 0.822i)9-s + 12-s − 0.709·13-s + (0.568 + 0.822i)16-s + (−1.32 − 1.17i)19-s + (−0.850 + 1.23i)21-s + (−0.970 + 0.239i)25-s + (0.120 − 0.992i)27-s − 1.49·28-s + (0.688 − 1.81i)31-s + (0.885 − 0.464i)36-s + (−0.180 + 1.48i)37-s + ⋯
L(s)  = 1  + (0.885 − 0.464i)3-s + (0.885 + 0.464i)4-s + (−1.32 + 0.695i)7-s + (0.568 − 0.822i)9-s + 12-s − 0.709·13-s + (0.568 + 0.822i)16-s + (−1.32 − 1.17i)19-s + (−0.850 + 1.23i)21-s + (−0.970 + 0.239i)25-s + (0.120 − 0.992i)27-s − 1.49·28-s + (0.688 − 1.81i)31-s + (0.885 − 0.464i)36-s + (−0.180 + 1.48i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.999 + 0.0195i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.999 + 0.0195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165336992\)
\(L(\frac12)\) \(\approx\) \(1.165336992\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.885 + 0.464i)T \)
157 \( 1 + (-0.885 + 0.464i)T \)
good2 \( 1 + (-0.885 - 0.464i)T^{2} \)
5 \( 1 + (0.970 - 0.239i)T^{2} \)
7 \( 1 + (1.32 - 0.695i)T + (0.568 - 0.822i)T^{2} \)
11 \( 1 + (-0.885 + 0.464i)T^{2} \)
13 \( 1 + 0.709T + T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + (1.32 + 1.17i)T + (0.120 + 0.992i)T^{2} \)
23 \( 1 + (0.354 - 0.935i)T^{2} \)
29 \( 1 + (0.970 + 0.239i)T^{2} \)
31 \( 1 + (-0.688 + 1.81i)T + (-0.748 - 0.663i)T^{2} \)
37 \( 1 + (0.180 - 1.48i)T + (-0.970 - 0.239i)T^{2} \)
41 \( 1 + (0.354 + 0.935i)T^{2} \)
43 \( 1 + (-1.00 - 0.527i)T + (0.568 + 0.822i)T^{2} \)
47 \( 1 + (0.970 - 0.239i)T^{2} \)
53 \( 1 + (-0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.120 + 0.992i)T^{2} \)
61 \( 1 + (-1.45 - 1.28i)T + (0.120 + 0.992i)T^{2} \)
67 \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \)
71 \( 1 + (-0.120 - 0.992i)T^{2} \)
73 \( 1 + (-0.213 + 0.112i)T + (0.568 - 0.822i)T^{2} \)
79 \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 + (0.748 + 0.663i)T^{2} \)
89 \( 1 + (0.748 + 0.663i)T^{2} \)
97 \( 1 + (-0.0290 - 0.239i)T + (-0.970 + 0.239i)T^{2} \)
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

−11.48459713219164186112225511259, −10.13671478407444443872722712600, −9.392830164131050700092357990761, −8.487257381850152077006487087024, −7.55609243208487077606672696112, −6.67567838392330942169048474505, −6.05476526679486829315426724634, −4.12421271250680396696453703744, −2.87725596750361358843168021063, −2.31426344277502362989468211581, 2.07756407480044758093342384301, 3.22545872380412369008781315465, 4.21348884437458485560700167817, 5.72563171485969535887054387504, 6.77045297811837712386358887874, 7.47606202611951228777534850173, 8.603662680495483171448682558770, 9.786536020702816039982019751801, 10.17399681785440376668063535457, 10.86699778345101054218624626276

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