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Properties

Label 2-471-157.144-c1-0-17
Degree $2$
Conductor $471$
Sign $-0.984 + 0.173i$
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  − 1.15·2-s + (−0.5 + 0.866i)3-s − 0.666·4-s + (1.08 − 1.87i)5-s + (0.577 − 0.999i)6-s − 1.15·7-s + 3.07·8-s + (−0.499 − 0.866i)9-s + (−1.24 + 2.16i)10-s + (−1.53 + 2.65i)11-s + (0.333 − 0.577i)12-s + (−1.04 − 1.80i)13-s + 1.33·14-s + (1.08 + 1.87i)15-s − 2.22·16-s + (−1.80 + 3.12i)17-s + ⋯
L(s)  = 1  − 0.816·2-s + (−0.288 + 0.499i)3-s − 0.333·4-s + (0.483 − 0.836i)5-s + (0.235 − 0.408i)6-s − 0.438·7-s + 1.08·8-s + (−0.166 − 0.288i)9-s + (−0.394 + 0.683i)10-s + (−0.461 + 0.800i)11-s + (0.0962 − 0.166i)12-s + (−0.289 − 0.500i)13-s + 0.357·14-s + (0.278 + 0.483i)15-s − 0.555·16-s + (−0.437 + 0.757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 471,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00269288 - 0.0308555i\)
\(L(\frac12)\) \(\approx\) \(0.00269288 - 0.0308555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (10.7 + 6.51i)T \)
good2 \( 1 + 1.15T + 2T^{2} \)
5 \( 1 + (-1.08 + 1.87i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.15T + 7T^{2} \)
11 \( 1 + (1.53 - 2.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.04 + 1.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.80 - 3.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 + 1.97i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 + 7.50T + 29T^{2} \)
31 \( 1 + (0.477 + 0.827i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.78 + 10.0i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 + (-1.08 - 1.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0102 - 0.0177i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.20 + 2.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.07T + 59T^{2} \)
61 \( 1 + (-0.567 + 0.983i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 3.29T + 67T^{2} \)
71 \( 1 + (-7.52 - 13.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.352 + 0.609i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.35T + 79T^{2} \)
83 \( 1 + (-6.03 - 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.62 - 2.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.32 + 4.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24927438920780496789924169850, −9.679640038528846790612855861934, −9.029049486508925907413539282369, −8.136964737971270907099303492402, −7.11310129352618461010137766114, −5.66059141490963043200600534949, −4.92190777879841257352229637988, −3.85037232675664038559840571679, −1.86301219605410580867071711502, −0.02516959990293157107593797014, 1.89668151782275080305787055139, 3.33530183923362148702935302546, 4.92894454402119113520741339335, 6.08462994044827115956590793544, 6.95806538927191599892255789753, 7.84257081502064548760885533724, 8.784734635288880877459823172077, 9.758479351869175365155522274463, 10.36210667206426773371441026273, 11.22366600761994694918577502267

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