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Properties

Label 2-471-157.156-c1-0-21
Degree $2$
Conductor $471$
Sign $0.409 + 0.912i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.560i·2-s + 3-s + 1.68·4-s + 0.365i·5-s − 0.560i·6-s − 4.51i·7-s − 2.06i·8-s + 9-s + 0.205·10-s + 2.17·11-s + 1.68·12-s − 6.15·13-s − 2.53·14-s + 0.365i·15-s + 2.21·16-s − 3.05·17-s + ⋯
L(s)  = 1  − 0.396i·2-s + 0.577·3-s + 0.842·4-s + 0.163i·5-s − 0.228i·6-s − 1.70i·7-s − 0.730i·8-s + 0.333·9-s + 0.0648·10-s + 0.655·11-s + 0.486·12-s − 1.70·13-s − 0.677·14-s + 0.0944i·15-s + 0.552·16-s − 0.741·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69396 - 1.09666i\)
\(L(\frac12)\) \(\approx\) \(1.69396 - 1.09666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
157 \( 1 + (-5.12 - 11.4i)T \)
good2 \( 1 + 0.560iT - 2T^{2} \)
5 \( 1 - 0.365iT - 5T^{2} \)
7 \( 1 + 4.51iT - 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 + 6.15T + 13T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 - 1.16T + 19T^{2} \)
23 \( 1 - 4.14iT - 23T^{2} \)
29 \( 1 - 7.44iT - 29T^{2} \)
31 \( 1 - 3.34T + 31T^{2} \)
37 \( 1 - 0.810T + 37T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + 2.50iT - 43T^{2} \)
47 \( 1 - 5.83T + 47T^{2} \)
53 \( 1 + 6.65iT - 53T^{2} \)
59 \( 1 - 6.16iT - 59T^{2} \)
61 \( 1 + 8.60iT - 61T^{2} \)
67 \( 1 + 5.54T + 67T^{2} \)
71 \( 1 - 7.49T + 71T^{2} \)
73 \( 1 - 5.54iT - 73T^{2} \)
79 \( 1 + 3.93iT - 79T^{2} \)
83 \( 1 - 5.95iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−10.78493848228018451844643637470, −10.07921427975413482421674228255, −9.362070463710816720356070234124, −7.892136303581423426595114214222, −7.08770523169626579168196289666, −6.71812151808644193231732446610, −4.82855658379284404384242061508, −3.75637172098514648838577182853, −2.73846133283067704412616512954, −1.29641843583300279974989113486, 2.21260031904393515084964687227, 2.73223263402201499981598810012, 4.60115857879832814482844607203, 5.66879341038245225809353354228, 6.60874762255017301177225833803, 7.51059238053807527349060984618, 8.547037351018848683332787848725, 9.155803049918354409812900724768, 10.14658500236040435870426488550, 11.37607487098105405042780309477

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