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Properties

Label 2-471-157.144-c1-0-7
Degree $2$
Conductor $471$
Sign $0.942 - 0.333i$
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  − 2.03·2-s + (−0.5 + 0.866i)3-s + 2.15·4-s + (−0.0788 + 0.136i)5-s + (1.01 − 1.76i)6-s − 4.73·7-s − 0.311·8-s + (−0.499 − 0.866i)9-s + (0.160 − 0.278i)10-s + (2.02 − 3.50i)11-s + (−1.07 + 1.86i)12-s + (1.05 + 1.81i)13-s + 9.65·14-s + (−0.0788 − 0.136i)15-s − 3.67·16-s + (1.09 − 1.89i)17-s + ⋯
L(s)  = 1  − 1.44·2-s + (−0.288 + 0.499i)3-s + 1.07·4-s + (−0.0352 + 0.0610i)5-s + (0.415 − 0.720i)6-s − 1.78·7-s − 0.110·8-s + (−0.166 − 0.288i)9-s + (0.0508 − 0.0880i)10-s + (0.610 − 1.05i)11-s + (−0.310 + 0.538i)12-s + (0.291 + 0.504i)13-s + 2.57·14-s + (−0.0203 − 0.0352i)15-s − 0.917·16-s + (0.265 − 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455419 + 0.0782728i\)
\(L(\frac12)\) \(\approx\) \(0.455419 + 0.0782728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
157 \( 1 + (-11.6 - 4.65i)T \)
good2 \( 1 + 2.03T + 2T^{2} \)
5 \( 1 + (0.0788 - 0.136i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.73T + 7T^{2} \)
11 \( 1 + (-2.02 + 3.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.05 - 1.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.09 + 1.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.198 - 0.343i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 + (1.77 + 3.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.209 + 0.362i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.59T + 41T^{2} \)
43 \( 1 + (-6.00 - 10.4i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.483 - 0.838i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.91 - 11.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.38T + 59T^{2} \)
61 \( 1 + (-7.13 + 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 6.81T + 67T^{2} \)
71 \( 1 + (3.75 + 6.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.51 - 2.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.64T + 79T^{2} \)
83 \( 1 + (7.27 + 12.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.47 - 4.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.81 - 4.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92812359514906354151094278234, −9.880306284849672506460916495555, −9.278042092312167480392351353358, −8.867120047214214615531629170988, −7.48028657230047633223326689448, −6.60265441845233282057759377299, −5.78818612239178891808041854080, −4.02881725784010853339748360050, −2.93198950640275974048251212689, −0.78230531827984030434788106037, 0.77928507738833266390137544641, 2.40720516592615035304506034786, 3.95116342639979819890302806259, 5.70102002082781351159367283442, 6.87124690341782767458649529389, 7.14372931903784547018196232561, 8.470021799696544717115021364214, 9.203637622601733431422145050492, 10.02697152206647643412316799656, 10.54405912321329895082527746716

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