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Properties

Label 2-471-157.144-c1-0-20
Degree $2$
Conductor $471$
Sign $-0.177 + 0.984i$
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  − 0.479·2-s + (0.5 − 0.866i)3-s − 1.76·4-s + (1.73 − 3.01i)5-s + (−0.239 + 0.415i)6-s + 2.28·7-s + 1.80·8-s + (−0.499 − 0.866i)9-s + (−0.834 + 1.44i)10-s + (−0.849 + 1.47i)11-s + (−0.884 + 1.53i)12-s + (−0.649 − 1.12i)13-s − 1.09·14-s + (−1.73 − 3.01i)15-s + 2.67·16-s + (3.76 − 6.52i)17-s + ⋯
L(s)  = 1  − 0.339·2-s + (0.288 − 0.499i)3-s − 0.884·4-s + (0.777 − 1.34i)5-s + (−0.0979 + 0.169i)6-s + 0.864·7-s + 0.639·8-s + (−0.166 − 0.288i)9-s + (−0.263 + 0.456i)10-s + (−0.256 + 0.443i)11-s + (−0.255 + 0.442i)12-s + (−0.180 − 0.311i)13-s − 0.293·14-s + (−0.448 − 0.777i)15-s + 0.668·16-s + (0.912 − 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.177 + 0.984i$
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: \((2,\ 471,\ (\ :1/2),\ -0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792586 - 0.948145i\)
\(L(\frac12)\) \(\approx\) \(0.792586 - 0.948145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-9.81 + 7.79i)T \)
good2 \( 1 + 0.479T + 2T^{2} \)
5 \( 1 + (-1.73 + 3.01i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.28T + 7T^{2} \)
11 \( 1 + (0.849 - 1.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.649 + 1.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.76 + 6.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.59 - 4.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.81T + 23T^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + (1.57 + 2.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.61 + 8.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 + (1.79 + 3.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.42 - 2.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 - 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.58T + 59T^{2} \)
61 \( 1 + (0.121 - 0.211i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 5.44T + 67T^{2} \)
71 \( 1 + (-1.38 - 2.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.92 - 5.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.23 + 2.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.54 - 9.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.41373510800020989082072568077, −9.687601501412067864940786636562, −8.954078766083507873097527733987, −8.117549381285733566523971592709, −7.58682360611423668087648822827, −5.77602834413189276943485893453, −5.09764576717476080950670039379, −4.15350783770772494532115480467, −2.10273253347020413098130476302, −0.902487291279285828003352444319, 1.93397830070430945868755451169, 3.34220281306574701844968688066, 4.48058770868289170715585625008, 5.59563531644381695222528237423, 6.62399285861509252962234573136, 8.008451243149543187134168923825, 8.483236131771701085959897791308, 9.673888061668965851692192783337, 10.38653700092022622429850398244, 10.74516368482167904035016103076

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