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Properties

Label 2-471-157.144-c1-0-18
Degree $2$
Conductor $471$
Sign $0.0980 + 0.995i$
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  + 0.537·2-s + (0.5 − 0.866i)3-s − 1.71·4-s + (−0.249 + 0.431i)5-s + (0.268 − 0.465i)6-s + 1.57·7-s − 1.99·8-s + (−0.499 − 0.866i)9-s + (−0.134 + 0.232i)10-s + (2.88 − 4.99i)11-s + (−0.855 + 1.48i)12-s + (−2.90 − 5.02i)13-s + 0.848·14-s + (0.249 + 0.431i)15-s + 2.34·16-s + (−0.250 + 0.433i)17-s + ⋯
L(s)  = 1  + 0.380·2-s + (0.288 − 0.499i)3-s − 0.855·4-s + (−0.111 + 0.193i)5-s + (0.109 − 0.190i)6-s + 0.596·7-s − 0.705·8-s + (−0.166 − 0.288i)9-s + (−0.0423 + 0.0734i)10-s + (0.869 − 1.50i)11-s + (−0.246 + 0.427i)12-s + (−0.805 − 1.39i)13-s + 0.226·14-s + (0.0643 + 0.111i)15-s + 0.587·16-s + (−0.0607 + 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.0980 + 0.995i$
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: \((2,\ 471,\ (\ :1/2),\ 0.0980 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06398 - 0.964331i\)
\(L(\frac12)\) \(\approx\) \(1.06398 - 0.964331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.31 + 10.1i)T \)
good2 \( 1 - 0.537T + 2T^{2} \)
5 \( 1 + (0.249 - 0.431i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + (-2.88 + 4.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.90 + 5.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.250 - 0.433i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.13 + 1.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 + 2.88T + 29T^{2} \)
31 \( 1 + (4.96 + 8.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.25 - 5.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.670T + 41T^{2} \)
43 \( 1 + (2.94 + 5.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.922 + 1.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.29 - 9.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.44T + 59T^{2} \)
61 \( 1 + (2.92 - 5.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 9.17T + 67T^{2} \)
71 \( 1 + (-6.95 - 12.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.00 - 5.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.23T + 79T^{2} \)
83 \( 1 + (-8.01 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.43 + 2.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.65 + 16.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.05243111548270153637425969983, −9.765392355597804588628895813461, −8.864566454178777661861034431726, −8.182843075073558422845592239079, −7.24670816106993851007085646307, −5.89962576688555030483255683228, −5.18517341618635048521203414643, −3.81747657352646502379420095915, −2.90020601268781641482625178427, −0.826437001509581221924586120753, 1.90010144945009161100985876012, 3.66699288568082426845988270188, 4.62814624776117835147953795673, 4.98666608069124587356646068366, 6.62168896902060766240168612324, 7.63162101294201076302617902918, 8.837998601512139482066016833579, 9.354529167888538548933011240024, 10.07741053133020018433559266383, 11.38027417518345308212657683402

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