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Properties

Label 2-471-157.156-c3-0-25
Degree $2$
Conductor $471$
Sign $0.371 - 0.928i$
Analytic cond. $27.7898$
Root an. cond. $5.27161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.46i·2-s − 3·3-s + 5.84·4-s + 4.94i·5-s − 4.40i·6-s − 12.8i·7-s + 20.3i·8-s + 9·9-s − 7.26·10-s − 22.3·11-s − 17.5·12-s + 72.1·13-s + 18.9·14-s − 14.8i·15-s + 16.8·16-s + 19.7·17-s + ⋯
L(s)  = 1  + 0.519i·2-s − 0.577·3-s + 0.730·4-s + 0.442i·5-s − 0.299i·6-s − 0.695i·7-s + 0.898i·8-s + 0.333·9-s − 0.229·10-s − 0.612·11-s − 0.421·12-s + 1.53·13-s + 0.361·14-s − 0.255i·15-s + 0.263·16-s + 0.282·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(27.7898\)
Root analytic conductor: \(5.27161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :3/2),\ 0.371 - 0.928i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.035717559\)
\(L(\frac12)\) \(\approx\) \(2.035717559\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
157 \( 1 + (-729. + 1.82e3i)T \)
good2 \( 1 - 1.46iT - 8T^{2} \)
5 \( 1 - 4.94iT - 125T^{2} \)
7 \( 1 + 12.8iT - 343T^{2} \)
11 \( 1 + 22.3T + 1.33e3T^{2} \)
13 \( 1 - 72.1T + 2.19e3T^{2} \)
17 \( 1 - 19.7T + 4.91e3T^{2} \)
19 \( 1 + 82.0T + 6.85e3T^{2} \)
23 \( 1 + 50.1iT - 1.21e4T^{2} \)
29 \( 1 - 147. iT - 2.43e4T^{2} \)
31 \( 1 - 144.T + 2.97e4T^{2} \)
37 \( 1 - 221.T + 5.06e4T^{2} \)
41 \( 1 + 22.4iT - 6.89e4T^{2} \)
43 \( 1 + 204. iT - 7.95e4T^{2} \)
47 \( 1 + 84.4T + 1.03e5T^{2} \)
53 \( 1 - 213. iT - 1.48e5T^{2} \)
59 \( 1 - 14.1iT - 2.05e5T^{2} \)
61 \( 1 - 811. iT - 2.26e5T^{2} \)
67 \( 1 - 19.0T + 3.00e5T^{2} \)
71 \( 1 - 217.T + 3.57e5T^{2} \)
73 \( 1 - 235. iT - 3.89e5T^{2} \)
79 \( 1 - 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.39e3T + 7.04e5T^{2} \)
97 \( 1 - 698. iT - 9.12e5T^{2} \)
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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

−10.67819305007609941312879925138, −10.40436477558386490094184315171, −8.726210474234148040903135677725, −7.86617824442804575594719856955, −6.86769691960001048392600707044, −6.33071914384011962795695948712, −5.35873545698618430592723794620, −4.02813544565669468634125839818, −2.68709722391621507260974666108, −1.11572514077144702664365929846, 0.830904393974451677888861736224, 2.07034758529824303706280333995, 3.33610026845878202530066124469, 4.64053310678904375612301647231, 5.92799088066000996652631755175, 6.39780365978112838882153967234, 7.77421711284282603866035174366, 8.660346023637592549080813087547, 9.773681112180649213146784846189, 10.68900564820313885272528222059

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