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Properties

Label 2-504-504.355-c1-0-41
Degree $2$
Conductor $504$
Sign $-0.853 - 0.520i$
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 + i)2-s + 1.73i·3-s + 2i·4-s + (0.866 + 1.5i)5-s + (−1.73 + 1.73i)6-s + (1.73 + 2i)7-s + (−2 + 2i)8-s − 2.99·9-s + (−0.633 + 2.36i)10-s + (2.5 − 4.33i)11-s − 3.46·12-s + (0.866 − 1.5i)13-s + (−0.267 + 3.73i)14-s + (−2.59 + 1.49i)15-s − 4·16-s + (1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 0.999i·3-s + i·4-s + (0.387 + 0.670i)5-s + (−0.707 + 0.707i)6-s + (0.654 + 0.755i)7-s + (−0.707 + 0.707i)8-s − 0.999·9-s + (−0.200 + 0.748i)10-s + (0.753 − 1.30i)11-s − 1.00·12-s + (0.240 − 0.416i)13-s + (−0.0716 + 0.997i)14-s + (−0.670 + 0.387i)15-s − 16-s + (0.363 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.610661 + 2.17476i\)
\(L(\frac12)\) \(\approx\) \(0.610661 + 2.17476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 + (-1.73 - 2i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.866 + 1.5i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.06 - 3.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.5 - 6.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 - 3.46T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-13.5 + 7.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11869377660800236781804366240, −10.76277471541343374099936443170, −9.061458436412093636200179357806, −8.839497247834557555156406670810, −7.64287110195845455600571975348, −6.30093137896911086764559836516, −5.71144359971032062863901817971, −4.79828964259493927283058134538, −3.55249505503508545433117141973, −2.72160982628742062208172612566, 1.27609857696236131307261105900, 1.94873601377448382464476098208, 3.71683860070410738945971181652, 4.78228879906470452476985319236, 5.71248863452681977192840752830, 6.85588749814962395249424105277, 7.60362909598465610715550719740, 9.025190350941085321027385135747, 9.620626581793326268685949025080, 10.98061195536271840498321492519

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