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Properties

Label 2-471-157.156-c3-0-67
Degree $2$
Conductor $471$
Sign $-0.918 - 0.396i$
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  − 1.38i·2-s − 3·3-s + 6.07·4-s + 4.30i·5-s + 4.16i·6-s − 36.7i·7-s − 19.5i·8-s + 9·9-s + 5.97·10-s − 64.0·11-s − 18.2·12-s + 37.9·13-s − 51.0·14-s − 12.9i·15-s + 21.4·16-s − 114.·17-s + ⋯
L(s)  = 1  − 0.491i·2-s − 0.577·3-s + 0.758·4-s + 0.384i·5-s + 0.283i·6-s − 1.98i·7-s − 0.863i·8-s + 0.333·9-s + 0.188·10-s − 1.75·11-s − 0.438·12-s + 0.809·13-s − 0.974·14-s − 0.222i·15-s + 0.334·16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.918 - 0.396i$
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: \((2,\ 471,\ (\ :3/2),\ -0.918 - 0.396i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7133039387\)
\(L(\frac12)\) \(\approx\) \(0.7133039387\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.80e3 + 779. i)T \)
good2 \( 1 + 1.38iT - 8T^{2} \)
5 \( 1 - 4.30iT - 125T^{2} \)
7 \( 1 + 36.7iT - 343T^{2} \)
11 \( 1 + 64.0T + 1.33e3T^{2} \)
13 \( 1 - 37.9T + 2.19e3T^{2} \)
17 \( 1 + 114.T + 4.91e3T^{2} \)
19 \( 1 - 66.8T + 6.85e3T^{2} \)
23 \( 1 - 90.8iT - 1.21e4T^{2} \)
29 \( 1 + 7.45iT - 2.43e4T^{2} \)
31 \( 1 + 236.T + 2.97e4T^{2} \)
37 \( 1 + 184.T + 5.06e4T^{2} \)
41 \( 1 + 11.4iT - 6.89e4T^{2} \)
43 \( 1 - 290. iT - 7.95e4T^{2} \)
47 \( 1 - 7.84T + 1.03e5T^{2} \)
53 \( 1 + 54.2iT - 1.48e5T^{2} \)
59 \( 1 + 457. iT - 2.05e5T^{2} \)
61 \( 1 + 410. iT - 2.26e5T^{2} \)
67 \( 1 + 161.T + 3.00e5T^{2} \)
71 \( 1 + 285.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 + 212. iT - 4.93e5T^{2} \)
83 \( 1 - 463. iT - 5.71e5T^{2} \)
89 \( 1 + 753.T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3iT - 9.12e5T^{2} \)
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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

−10.59276680072322727417102088316, −9.671302032172941273490286055985, −7.965468568941431918853339284555, −7.16670866358808968637333468273, −6.64214660144525525191801206672, −5.28809679786606744710436485493, −4.04276096553764585301477743185, −3.04007387234833677492309074913, −1.52372649728862951418372617847, −0.22281515800580913379450675352, 1.99189591573695649570055336890, 2.85426035816447051590349832086, 4.98310479016637495901031341258, 5.52960618652973718893702578503, 6.28837815989437142948692383683, 7.36211727880362737711617987704, 8.524015243506064201777597861660, 8.944025500345142748131241002221, 10.50974085473466616593180171908, 11.10306495698723635801034094253

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