Properties

Label 2-471-157.156-c1-0-11
Degree $2$
Conductor $471$
Sign $0.828 - 0.560i$
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  + 0.430i·2-s − 3-s + 1.81·4-s + 1.89i·5-s − 0.430i·6-s − 3.40i·7-s + 1.64i·8-s + 9-s − 0.815·10-s + 2.33·11-s − 1.81·12-s + 1.46·13-s + 1.46·14-s − 1.89i·15-s + 2.92·16-s + 5.58·17-s + ⋯
L(s)  = 1  + 0.304i·2-s − 0.577·3-s + 0.907·4-s + 0.847i·5-s − 0.175i·6-s − 1.28i·7-s + 0.580i·8-s + 0.333·9-s − 0.257·10-s + 0.703·11-s − 0.523·12-s + 0.405·13-s + 0.391·14-s − 0.489i·15-s + 0.731·16-s + 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.828 - 0.560i$
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: \((2,\ 471,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48691 + 0.455687i\)
\(L(\frac12)\) \(\approx\) \(1.48691 + 0.455687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (10.3 - 7.02i)T \)
good2 \( 1 - 0.430iT - 2T^{2} \)
5 \( 1 - 1.89iT - 5T^{2} \)
7 \( 1 + 3.40iT - 7T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 - 5.58T + 17T^{2} \)
19 \( 1 + 6.04T + 19T^{2} \)
23 \( 1 - 8.56iT - 23T^{2} \)
29 \( 1 + 8.19iT - 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 - 3.26iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 - 1.67T + 47T^{2} \)
53 \( 1 + 0.511iT - 53T^{2} \)
59 \( 1 + 9.95iT - 59T^{2} \)
61 \( 1 + 1.38iT - 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 16.3iT - 73T^{2} \)
79 \( 1 + 1.85iT - 79T^{2} \)
83 \( 1 - 3.55iT - 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−11.17368705767769496743363180133, −10.40982947977186338944384632390, −9.639575116421641971697370307612, −7.88734337678687938745714547762, −7.37599766364338137776876537679, −6.43827123617093046638420519679, −5.88419405216820050661670081827, −4.24811210790501722232640929321, −3.19023641857234563727825434211, −1.44104940095751852713150830469, 1.29491582112469339467868336614, 2.61776995217320834797187931268, 4.12508580286534252360239450144, 5.44917025251564198018089750923, 6.11780822630854692766725200942, 7.08176709351400347881411599800, 8.478610308636150788681484704470, 9.031679013832578378579704961070, 10.29915603375029664160725912498, 10.98121403186153826754601357417

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