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Properties

Label 2-471-157.156-c1-0-9
Degree $2$
Conductor $471$
Sign $-0.515 - 0.856i$
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  + 1.66i·2-s + 3-s − 0.764·4-s + 1.85i·5-s + 1.66i·6-s − 0.948i·7-s + 2.05i·8-s + 9-s − 3.08·10-s + 5.18·11-s − 0.764·12-s + 2.15·13-s + 1.57·14-s + 1.85i·15-s − 4.94·16-s − 1.84·17-s + ⋯
L(s)  = 1  + 1.17i·2-s + 0.577·3-s − 0.382·4-s + 0.829i·5-s + 0.678i·6-s − 0.358i·7-s + 0.726i·8-s + 0.333·9-s − 0.975·10-s + 1.56·11-s − 0.220·12-s + 0.596·13-s + 0.421·14-s + 0.479i·15-s − 1.23·16-s − 0.447·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.515 - 0.856i$
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: \((2,\ 471,\ (\ :1/2),\ -0.515 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930134 + 1.64483i\)
\(L(\frac12)\) \(\approx\) \(0.930134 + 1.64483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.45 + 10.7i)T \)
good2 \( 1 - 1.66iT - 2T^{2} \)
5 \( 1 - 1.85iT - 5T^{2} \)
7 \( 1 + 0.948iT - 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + 8.04T + 19T^{2} \)
23 \( 1 + 4.20iT - 23T^{2} \)
29 \( 1 - 5.35iT - 29T^{2} \)
31 \( 1 + 2.86T + 31T^{2} \)
37 \( 1 + 1.28T + 37T^{2} \)
41 \( 1 + 0.936iT - 41T^{2} \)
43 \( 1 - 0.0513iT - 43T^{2} \)
47 \( 1 - 1.22T + 47T^{2} \)
53 \( 1 - 0.958iT - 53T^{2} \)
59 \( 1 + 3.20iT - 59T^{2} \)
61 \( 1 + 13.5iT - 61T^{2} \)
67 \( 1 + 3.65T + 67T^{2} \)
71 \( 1 - 2.98T + 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + 3.34iT - 79T^{2} \)
83 \( 1 + 15.2iT - 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 18.3iT - 97T^{2} \)
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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

−11.03449700436833458354603268212, −10.57024971303500260591799879951, −9.026672202108386632704681589709, −8.626194283732260086181387582004, −7.47189326094938018768432194472, −6.55140269205892692260676258447, −6.36575351741975145338722180711, −4.62658108045802372279424975994, −3.57387021890502815811450423309, −2.06516555753011271957889006811, 1.26277723367651899830282897069, 2.30906469298117825203646408499, 3.77051689098170286698062805258, 4.35780879096633668133529388318, 6.04798049238562634520140069795, 7.01922284929065858259055356932, 8.573921402351491387021685015916, 8.960352504149960305789198882733, 9.796131052278285370156749102832, 10.85878229106152261801878811115

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