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Properties

Label 2-504-56.3-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.489 - 0.871i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.35 + 0.406i)2-s + (1.66 − 1.10i)4-s + (1.09 + 1.89i)5-s + (1.40 + 2.24i)7-s + (−1.81 + 2.16i)8-s + (−2.24 − 2.11i)10-s + (−2.14 + 3.71i)11-s − 2.03·13-s + (−2.80 − 2.47i)14-s + (1.57 − 3.67i)16-s + (−1.56 − 0.903i)17-s + (0.509 − 0.294i)19-s + (3.90 + 1.95i)20-s + (1.39 − 5.90i)22-s + (0.146 − 0.0846i)23-s + ⋯
L(s)  = 1  + (−0.957 + 0.287i)2-s + (0.834 − 0.550i)4-s + (0.488 + 0.845i)5-s + (0.529 + 0.848i)7-s + (−0.641 + 0.767i)8-s + (−0.710 − 0.669i)10-s + (−0.646 + 1.11i)11-s − 0.564·13-s + (−0.750 − 0.660i)14-s + (0.394 − 0.919i)16-s + (−0.379 − 0.219i)17-s + (0.116 − 0.0674i)19-s + (0.873 + 0.437i)20-s + (0.297 − 1.25i)22-s + (0.0305 − 0.0176i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.489 - 0.871i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.489 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437658 + 0.747715i\)
\(L(\frac12)\) \(\approx\) \(0.437658 + 0.747715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.406i)T \)
3 \( 1 \)
7 \( 1 + (-1.40 - 2.24i)T \)
good5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.14 - 3.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.03T + 13T^{2} \)
17 \( 1 + (1.56 + 0.903i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.509 + 0.294i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.146 + 0.0846i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.439iT - 29T^{2} \)
31 \( 1 + (2.82 - 4.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.72 - 4.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 - 8.59T + 43T^{2} \)
47 \( 1 + (-5.47 - 9.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.53 + 2.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.730 - 0.421i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.22 - 7.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.87 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.72iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.784 + 0.453i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.4iT - 83T^{2} \)
89 \( 1 + (-14.5 + 8.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.3iT - 97T^{2} \)
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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

−10.84397913660376943620016489156, −10.32018704847586154131986580111, −9.417379912924189438445174541894, −8.638869381311222179348177189789, −7.49718405906803739208788706426, −6.92918304007607286294653727885, −5.79364035671882240289797302929, −4.89890444382505799636192021078, −2.73442530598091653099728909761, −1.98359638527918453407871338754, 0.68294082518804047375237920977, 2.06164149651930114127098439077, 3.57197448669846265594703821412, 4.95608027381396233188310198077, 6.05796219323900161364654955672, 7.31798186406511177675435630071, 8.068802594509642321845871456437, 8.881005415421064480660128479948, 9.680667409569217762986378272705, 10.66171903489799143928241509376

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