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Properties

Label 2-476-7.4-c1-0-6
Degree $2$
Conductor $476$
Sign $0.605 + 0.795i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
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  + (0.5 − 0.866i)3-s + (2 − 1.73i)7-s + (1 + 1.73i)9-s + (2.5 − 4.33i)11-s − 5·13-s + (0.5 − 0.866i)17-s + (3 + 5.19i)19-s + (−0.499 − 2.59i)21-s + (−2 − 3.46i)23-s + (2.5 − 4.33i)25-s + 5·27-s + 4·29-s + (−2.5 − 4.33i)33-s + (−4 − 6.92i)37-s + (−2.5 + 4.33i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (0.753 − 1.30i)11-s − 1.38·13-s + (0.121 − 0.210i)17-s + (0.688 + 1.19i)19-s + (−0.109 − 0.566i)21-s + (−0.417 − 0.722i)23-s + (0.5 − 0.866i)25-s + 0.962·27-s + 0.742·29-s + (−0.435 − 0.753i)33-s + (−0.657 − 1.13i)37-s + (−0.400 + 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51781 - 0.752422i\)
\(L(\frac12)\) \(\approx\) \(1.51781 - 0.752422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−10.76070078927853563800227546472, −10.16665789367693681511386836437, −8.920383449795084278142472428340, −7.979761100353735473653471487696, −7.43828929365060795735738802250, −6.34026231463760749894498317388, −5.09396971022954596103606947524, −4.07215862131710558042867532441, −2.61363896895923577898072564749, −1.17778144460264655931512169754, 1.76410491695041619740276124733, 3.16590876743410920438074084037, 4.57593435215229523296921584938, 5.09884622381532029711602365656, 6.70081315645900508324114976890, 7.41990891745679063868411709760, 8.615433627680208873560939957816, 9.581463170252919500187415214443, 9.835149806812365384822016159906, 11.23531069903192166498734560927

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