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Properties

Label 2-471-471.287-c0-0-0
Degree $2$
Conductor $471$
Sign $0.957 - 0.288i$
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  + (0.568 + 0.822i)3-s + (0.568 − 0.822i)4-s + (0.136 + 0.198i)7-s + (−0.354 + 0.935i)9-s + 12-s − 1.49·13-s + (−0.354 − 0.935i)16-s + (0.136 − 1.12i)19-s + (−0.0854 + 0.225i)21-s + (0.885 + 0.464i)25-s + (−0.970 + 0.239i)27-s + 0.241·28-s + (−1.32 + 1.17i)31-s + (0.568 + 0.822i)36-s + (−0.234 + 0.0576i)37-s + ⋯
L(s)  = 1  + (0.568 + 0.822i)3-s + (0.568 − 0.822i)4-s + (0.136 + 0.198i)7-s + (−0.354 + 0.935i)9-s + 12-s − 1.49·13-s + (−0.354 − 0.935i)16-s + (0.136 − 1.12i)19-s + (−0.0854 + 0.225i)21-s + (0.885 + 0.464i)25-s + (−0.970 + 0.239i)27-s + 0.241·28-s + (−1.32 + 1.17i)31-s + (0.568 + 0.822i)36-s + (−0.234 + 0.0576i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.957 - 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089605311\)
\(L(\frac12)\) \(\approx\) \(1.089605311\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.568 - 0.822i)T \)
157 \( 1 + (-0.568 - 0.822i)T \)
good2 \( 1 + (-0.568 + 0.822i)T^{2} \)
5 \( 1 + (-0.885 - 0.464i)T^{2} \)
7 \( 1 + (-0.136 - 0.198i)T + (-0.354 + 0.935i)T^{2} \)
11 \( 1 + (-0.568 - 0.822i)T^{2} \)
13 \( 1 + 1.49T + T^{2} \)
17 \( 1 + (-0.885 - 0.464i)T^{2} \)
19 \( 1 + (-0.136 + 1.12i)T + (-0.970 - 0.239i)T^{2} \)
23 \( 1 + (0.748 - 0.663i)T^{2} \)
29 \( 1 + (-0.885 + 0.464i)T^{2} \)
31 \( 1 + (1.32 - 1.17i)T + (0.120 - 0.992i)T^{2} \)
37 \( 1 + (0.234 - 0.0576i)T + (0.885 - 0.464i)T^{2} \)
41 \( 1 + (0.748 + 0.663i)T^{2} \)
43 \( 1 + (0.402 - 0.583i)T + (-0.354 - 0.935i)T^{2} \)
47 \( 1 + (-0.885 - 0.464i)T^{2} \)
53 \( 1 + (0.970 + 0.239i)T^{2} \)
59 \( 1 + (0.970 - 0.239i)T^{2} \)
61 \( 1 + (-0.213 + 1.75i)T + (-0.970 - 0.239i)T^{2} \)
67 \( 1 + (0.0854 - 0.704i)T + (-0.970 - 0.239i)T^{2} \)
71 \( 1 + (0.970 + 0.239i)T^{2} \)
73 \( 1 + (1.10 + 1.59i)T + (-0.354 + 0.935i)T^{2} \)
79 \( 1 + (-1.00 - 0.527i)T + (0.568 + 0.822i)T^{2} \)
83 \( 1 + (-0.120 + 0.992i)T^{2} \)
89 \( 1 + (-0.120 + 0.992i)T^{2} \)
97 \( 1 + (-1.88 - 0.464i)T + (0.885 + 0.464i)T^{2} \)
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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

−11.03664508054226005999261285401, −10.37940938322992024288173223398, −9.513745472438732505576710575358, −8.899917994107760076625761942361, −7.57612119738571074960022157051, −6.76295839055449735939762832634, −5.25925046505210213420483242628, −4.84936020574830460867940879108, −3.16249719392977288788608261356, −2.11560295727116135558741535638, 1.97842388995335004174107290122, 2.98554184768445309752082096276, 4.15845729518470631517085319446, 5.76641185431389128756927724371, 6.97424770632536482626684187279, 7.48924436292122013335784420656, 8.253673658701639095302455200687, 9.229163957285191648262191407601, 10.31558454105998288024903102212, 11.46386787673213865139804929636

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