Properties

Label 2-471-471.203-c0-0-0
Degree $2$
Conductor $471$
Sign $0.968 - 0.247i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.970 − 0.239i)3-s + (−0.970 + 0.239i)4-s + (0.688 + 0.169i)7-s + (0.885 + 0.464i)9-s + 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (0.688 + 1.81i)19-s + (−0.627 − 0.329i)21-s + (0.120 − 0.992i)25-s + (−0.748 − 0.663i)27-s − 0.709·28-s + (0.136 − 0.198i)31-s + (−0.970 − 0.239i)36-s + (0.530 + 0.470i)37-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)3-s + (−0.970 + 0.239i)4-s + (0.688 + 0.169i)7-s + (0.885 + 0.464i)9-s + 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (0.688 + 1.81i)19-s + (−0.627 − 0.329i)21-s + (0.120 − 0.992i)25-s + (−0.748 − 0.663i)27-s − 0.709·28-s + (0.136 − 0.198i)31-s + (−0.970 − 0.239i)36-s + (0.530 + 0.470i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.968 - 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5990915180\)
\(L(\frac12)\) \(\approx\) \(0.5990915180\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.970 + 0.239i)T \)
157 \( 1 + (0.970 + 0.239i)T \)
good2 \( 1 + (0.970 - 0.239i)T^{2} \)
5 \( 1 + (-0.120 + 0.992i)T^{2} \)
7 \( 1 + (-0.688 - 0.169i)T + (0.885 + 0.464i)T^{2} \)
11 \( 1 + (0.970 + 0.239i)T^{2} \)
13 \( 1 - 1.13T + T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (-0.688 - 1.81i)T + (-0.748 + 0.663i)T^{2} \)
23 \( 1 + (-0.568 + 0.822i)T^{2} \)
29 \( 1 + (-0.120 - 0.992i)T^{2} \)
31 \( 1 + (-0.136 + 0.198i)T + (-0.354 - 0.935i)T^{2} \)
37 \( 1 + (-0.530 - 0.470i)T + (0.120 + 0.992i)T^{2} \)
41 \( 1 + (-0.568 - 0.822i)T^{2} \)
43 \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \)
47 \( 1 + (-0.120 + 0.992i)T^{2} \)
53 \( 1 + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.748 + 0.663i)T^{2} \)
61 \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.748 - 0.663i)T^{2} \)
73 \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \)
79 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
83 \( 1 + (0.354 + 0.935i)T^{2} \)
89 \( 1 + (0.354 + 0.935i)T^{2} \)
97 \( 1 + (-1.12 + 0.992i)T + (0.120 - 0.992i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.40202510525128431109084392717, −10.38517078186039414378713373241, −9.648884664552922452709263200555, −8.309100733531715976110334991340, −7.913737686450859830688819177721, −6.44088166338679659306268401156, −5.56704543950275113376817637692, −4.67648887356512897772978260441, −3.64437851790998170883666325084, −1.42150823414168127023050983721, 1.17192534799501248123596547978, 3.60061232732325838989883284059, 4.72162481967863400101964123165, 5.29752581786521150475686963854, 6.39738859850787641843489264959, 7.51916190209366629766470698597, 8.726108187957457382425562928880, 9.413067893279275757393960107001, 10.45242532939195674984552998647, 11.16725412741099258873232953985

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