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Properties

Label 2-471-157.156-c3-0-56
Degree $2$
Conductor $471$
Sign $0.520 + 0.854i$
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  + 0.999i·2-s − 3·3-s + 7.00·4-s − 10.0i·5-s − 2.99i·6-s − 4.69i·7-s + 14.9i·8-s + 9·9-s + 10.0·10-s − 16.1·11-s − 21.0·12-s − 19.3·13-s + 4.69·14-s + 30.1i·15-s + 41.0·16-s + 117.·17-s + ⋯
L(s)  = 1  + 0.353i·2-s − 0.577·3-s + 0.875·4-s − 0.897i·5-s − 0.203i·6-s − 0.253i·7-s + 0.662i·8-s + 0.333·9-s + 0.317·10-s − 0.443·11-s − 0.505·12-s − 0.413·13-s + 0.0895·14-s + 0.518i·15-s + 0.641·16-s + 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.520 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.520 + 0.854i$
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: \((2,\ 471,\ (\ :3/2),\ 0.520 + 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.794293572\)
\(L(\frac12)\) \(\approx\) \(1.794293572\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.02e3 - 1.68e3i)T \)
good2 \( 1 - 0.999iT - 8T^{2} \)
5 \( 1 + 10.0iT - 125T^{2} \)
7 \( 1 + 4.69iT - 343T^{2} \)
11 \( 1 + 16.1T + 1.33e3T^{2} \)
13 \( 1 + 19.3T + 2.19e3T^{2} \)
17 \( 1 - 117.T + 4.91e3T^{2} \)
19 \( 1 - 14.0T + 6.85e3T^{2} \)
23 \( 1 + 69.8iT - 1.21e4T^{2} \)
29 \( 1 + 142. iT - 2.43e4T^{2} \)
31 \( 1 + 279.T + 2.97e4T^{2} \)
37 \( 1 + 239.T + 5.06e4T^{2} \)
41 \( 1 + 209. iT - 6.89e4T^{2} \)
43 \( 1 + 127. iT - 7.95e4T^{2} \)
47 \( 1 - 185.T + 1.03e5T^{2} \)
53 \( 1 + 746. iT - 1.48e5T^{2} \)
59 \( 1 + 590. iT - 2.05e5T^{2} \)
61 \( 1 + 530. iT - 2.26e5T^{2} \)
67 \( 1 - 365.T + 3.00e5T^{2} \)
71 \( 1 - 161.T + 3.57e5T^{2} \)
73 \( 1 - 33.4iT - 3.89e5T^{2} \)
79 \( 1 - 602. iT - 4.93e5T^{2} \)
83 \( 1 - 697. iT - 5.71e5T^{2} \)
89 \( 1 - 394.T + 7.04e5T^{2} \)
97 \( 1 - 909. iT - 9.12e5T^{2} \)
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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

−10.51296003812327837547615548444, −9.716654655594804479089496790570, −8.439296373859062710492777644980, −7.60622935207764970343345856908, −6.81103201407112975080584767849, −5.56836408417113075438369551169, −5.13876586832332134296624704959, −3.59384714133291722023771081302, −2.00989909646267997347254100982, −0.63306278289963273529572645495, 1.32603225889382002281897882182, 2.70361881577319931660340932888, 3.55813522031395960305982461285, 5.30111985122059797979664554078, 6.04122086425610166599770608649, 7.23138597001922296647689484543, 7.54763988063492005326235427994, 9.211229583608101069117202384812, 10.43118144772180915671146346868, 10.53833004145762577883688895767

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