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Properties

Label 2-504-168.107-c1-0-15
Degree $2$
Conductor $504$
Sign $0.975 - 0.221i$
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  + (−0.352 + 1.36i)2-s + (−1.75 − 0.965i)4-s + (0.635 − 1.09i)5-s + (−2.64 − 0.106i)7-s + (1.93 − 2.05i)8-s + (1.28 + 1.25i)10-s + (1.05 − 0.606i)11-s + 3.91i·13-s + (1.07 − 3.58i)14-s + (2.13 + 3.38i)16-s + (5.05 − 2.91i)17-s + (3.80 − 6.58i)19-s + (−2.17 + 1.31i)20-s + (0.460 + 1.65i)22-s + (4.20 − 7.28i)23-s + ⋯
L(s)  = 1  + (−0.249 + 0.968i)2-s + (−0.875 − 0.482i)4-s + (0.284 − 0.491i)5-s + (−0.999 − 0.0401i)7-s + (0.685 − 0.727i)8-s + (0.405 + 0.397i)10-s + (0.316 − 0.182i)11-s + 1.08i·13-s + (0.287 − 0.957i)14-s + (0.534 + 0.845i)16-s + (1.22 − 0.707i)17-s + (0.872 − 1.51i)19-s + (−0.486 + 0.293i)20-s + (0.0981 + 0.352i)22-s + (0.877 − 1.51i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10879 + 0.124154i\)
\(L(\frac12)\) \(\approx\) \(1.10879 + 0.124154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.352 - 1.36i)T \)
3 \( 1 \)
7 \( 1 + (2.64 + 0.106i)T \)
good5 \( 1 + (-0.635 + 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.05 + 0.606i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.91iT - 13T^{2} \)
17 \( 1 + (-5.05 + 2.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.80 + 6.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.20 + 7.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
31 \( 1 + (-2.14 + 1.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.24 + 3.02i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.01iT - 41T^{2} \)
43 \( 1 + 7.49T + 43T^{2} \)
47 \( 1 + (0.704 - 1.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.74 + 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.86 - 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.16 + 2.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.22 + 3.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + (-6.19 - 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.204 + 0.117i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (-8.38 - 4.84i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.07T + 97T^{2} \)
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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

−10.71445872448668670029482034200, −9.557268521102996816771803356523, −9.281919592714530105845179194658, −8.353648637442717032130606436377, −6.97355417604209059409261567210, −6.66186314470850068701018180262, −5.37230654137311785738430414812, −4.58154777742486034921178359761, −3.10531436812880383266398882983, −0.880989180176753525274834722302, 1.33612143034724269802665864530, 3.10665044546936588396652237987, 3.48805035971312352895596215625, 5.18999222112512292608738731034, 6.16244134424072799714498474294, 7.46685049790179074616885897975, 8.318185840815838407091120532502, 9.489979041166029021949561730440, 10.17089913834653609152353232285, 10.52621777342813662422004240738

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