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Properties

Label 2-504-56.3-c1-0-10
Degree $2$
Conductor $504$
Sign $-0.481 - 0.876i$
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  + (1.03 + 0.967i)2-s + (0.128 + 1.99i)4-s + (1.25 + 2.16i)5-s + (−1.36 − 2.26i)7-s + (−1.79 + 2.18i)8-s + (−0.805 + 3.44i)10-s + (−2.83 + 4.91i)11-s + 5.31·13-s + (0.787 − 3.65i)14-s + (−3.96 + 0.512i)16-s + (0.393 + 0.227i)17-s + (−3.19 + 1.84i)19-s + (−4.16 + 2.77i)20-s + (−7.68 + 2.32i)22-s + (4.43 − 2.56i)23-s + ⋯
L(s)  = 1  + (0.729 + 0.684i)2-s + (0.0642 + 0.997i)4-s + (0.559 + 0.969i)5-s + (−0.515 − 0.857i)7-s + (−0.635 + 0.771i)8-s + (−0.254 + 1.08i)10-s + (−0.855 + 1.48i)11-s + 1.47·13-s + (0.210 − 0.977i)14-s + (−0.991 + 0.128i)16-s + (0.0955 + 0.0551i)17-s + (−0.733 + 0.423i)19-s + (−0.931 + 0.620i)20-s + (−1.63 + 0.495i)22-s + (0.924 − 0.533i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.481 - 0.876i$
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: \((2,\ 504,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04775 + 1.77170i\)
\(L(\frac12)\) \(\approx\) \(1.04775 + 1.77170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 0.967i)T \)
3 \( 1 \)
7 \( 1 + (1.36 + 2.26i)T \)
good5 \( 1 + (-1.25 - 2.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.83 - 4.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.31T + 13T^{2} \)
17 \( 1 + (-0.393 - 0.227i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.19 - 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.43 + 2.56i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.57iT - 29T^{2} \)
31 \( 1 + (3.00 - 5.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.80 + 4.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + (0.478 + 0.829i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.41 + 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.76 - 5.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.50 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.65 + 8.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.35iT - 71T^{2} \)
73 \( 1 + (-5.93 - 3.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.71 + 4.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.96iT - 83T^{2} \)
89 \( 1 + (5.91 - 3.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.71iT - 97T^{2} \)
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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

−10.96187308260191485602840554292, −10.55205046510302955237478994153, −9.485596838488724851366088044894, −8.240533689915239928712945573041, −7.23008481298840059290307684002, −6.66286277413492956565587191765, −5.81651705249047280724177460064, −4.52380861828157371832263231451, −3.53683862770110482387139612489, −2.34491598659545758488739436854, 1.03743484733497864938880000891, 2.60236527999040969419776516149, 3.62526309017378015666342258040, 5.06200901250755535328526285907, 5.75125452952876561401793646283, 6.38474947293369021587323045946, 8.316565444408763433391789373285, 8.981636992447223246258457447359, 9.715907825906665393087065601483, 11.03702929722915991895814706971

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