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Properties

Label 2-471-157.144-c1-0-2
Degree $2$
Conductor $471$
Sign $-0.198 - 0.980i$
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  − 0.685·2-s + (0.5 − 0.866i)3-s − 1.53·4-s + (−0.295 + 0.512i)5-s + (−0.342 + 0.593i)6-s − 1.98·7-s + 2.41·8-s + (−0.499 − 0.866i)9-s + (0.202 − 0.351i)10-s + (0.0905 − 0.156i)11-s + (−0.765 + 1.32i)12-s + (0.547 + 0.948i)13-s + 1.36·14-s + (0.295 + 0.512i)15-s + 1.40·16-s + (−2.67 + 4.63i)17-s + ⋯
L(s)  = 1  − 0.484·2-s + (0.288 − 0.499i)3-s − 0.765·4-s + (−0.132 + 0.228i)5-s + (−0.139 + 0.242i)6-s − 0.751·7-s + 0.855·8-s + (−0.166 − 0.288i)9-s + (0.0640 − 0.111i)10-s + (0.0272 − 0.0472i)11-s + (−0.220 + 0.382i)12-s + (0.151 + 0.263i)13-s + 0.364·14-s + (0.0763 + 0.132i)15-s + 0.350·16-s + (−0.648 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 471,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299553 + 0.366321i\)
\(L(\frac12)\) \(\approx\) \(0.299553 + 0.366321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.24 - 10.8i)T \)
good2 \( 1 + 0.685T + 2T^{2} \)
5 \( 1 + (0.295 - 0.512i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.98T + 7T^{2} \)
11 \( 1 + (-0.0905 + 0.156i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.547 - 0.948i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.67 - 4.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.51 - 4.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + (-3.74 - 6.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.38 - 4.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.27 - 2.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.94T + 59T^{2} \)
61 \( 1 + (-2.48 + 4.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 + (6.09 + 10.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.82 - 8.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.82T + 79T^{2} \)
83 \( 1 + (1.34 + 2.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.46 - 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.64 + 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06989072946832987394672053387, −10.22993204822140649862644106288, −9.393449277076674665304128060165, −8.557276879278959873261952194548, −7.86892253673089398012212218489, −6.74915571447143654646258683924, −5.86247153149173881142030143499, −4.33409019153685863467071285265, −3.36745621093911370702006847345, −1.64074691149548718135413731672, 0.33911416659228978009775446307, 2.62171809218633418302664072006, 4.04360639548475778874977830043, 4.76567462503234776010221254738, 6.05522039702250258004831987912, 7.33764919570881583925883573901, 8.292060229329019337832136182407, 9.187400895272844104168942731419, 9.587957078993514742340664743241, 10.54466383085744100571819536438

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