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Properties

Label 2-471-157.144-c1-0-21
Degree $2$
Conductor $471$
Sign $-0.855 + 0.518i$
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  − 2.18·2-s + (0.5 − 0.866i)3-s + 2.75·4-s + (1.44 − 2.50i)5-s + (−1.09 + 1.88i)6-s − 2.39·7-s − 1.64·8-s + (−0.499 − 0.866i)9-s + (−3.15 + 5.45i)10-s + (1.50 − 2.60i)11-s + (1.37 − 2.38i)12-s + (−1.85 − 3.21i)13-s + 5.23·14-s + (−1.44 − 2.50i)15-s − 1.92·16-s + (0.282 − 0.489i)17-s + ⋯
L(s)  = 1  − 1.54·2-s + (0.288 − 0.499i)3-s + 1.37·4-s + (0.646 − 1.11i)5-s + (−0.445 + 0.770i)6-s − 0.906·7-s − 0.582·8-s + (−0.166 − 0.288i)9-s + (−0.996 + 1.72i)10-s + (0.453 − 0.785i)11-s + (0.397 − 0.688i)12-s + (−0.515 − 0.892i)13-s + 1.39·14-s + (−0.373 − 0.646i)15-s − 0.480·16-s + (0.0684 − 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.855 + 0.518i$
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: \((2,\ 471,\ (\ :1/2),\ -0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150526 - 0.538960i\)
\(L(\frac12)\) \(\approx\) \(0.150526 - 0.538960i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-12.3 - 2.19i)T \)
good2 \( 1 + 2.18T + 2T^{2} \)
5 \( 1 + (-1.44 + 2.50i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.39T + 7T^{2} \)
11 \( 1 + (-1.50 + 2.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.85 + 3.21i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.282 + 0.489i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 - 2.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.26T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (-2.06 - 3.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.400 + 0.694i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.20T + 41T^{2} \)
43 \( 1 + (2.18 + 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.83 + 6.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.24 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + (0.833 - 1.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 4.61T + 67T^{2} \)
71 \( 1 + (-6.39 - 11.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.04 - 3.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.08T + 79T^{2} \)
83 \( 1 + (2.73 + 4.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.21 + 9.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12030851811191401720516959078, −9.719477165289685511339498519472, −8.649734615502165639457678513829, −8.447587272905235187544767526370, −7.19399726475480809044311733921, −6.30183557398310709110911377400, −5.16996904851377251288643286641, −3.26352818363723501115278355586, −1.75398607079632187949677810532, −0.53637247464789857558548553275, 1.98723904282254447940706864167, 3.02053838364157263178186474173, 4.58877932194099697864139472078, 6.46490061042402809411375110020, 6.81411921058160488756869371137, 7.85912944988375818127950415631, 9.259226286958291789332495149881, 9.420984130053362009199704393470, 10.23265788139172918212140496636, 10.87865525240314272729901866933

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