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Properties

Label 2-504-63.16-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.913 + 0.406i$
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  + (0.677 + 1.59i)3-s − 2.66·5-s + (−0.654 − 2.56i)7-s + (−2.08 + 2.15i)9-s − 3.98·11-s + (1.00 + 1.73i)13-s + (−1.80 − 4.25i)15-s + (−3.57 − 6.18i)17-s + (−4.01 + 6.96i)19-s + (3.64 − 2.78i)21-s − 0.887·23-s + 2.12·25-s + (−4.85 − 1.85i)27-s + (−1.35 + 2.33i)29-s + (0.614 − 1.06i)31-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)3-s − 1.19·5-s + (−0.247 − 0.968i)7-s + (−0.694 + 0.719i)9-s − 1.20·11-s + (0.277 + 0.480i)13-s + (−0.466 − 1.09i)15-s + (−0.866 − 1.50i)17-s + (−0.922 + 1.59i)19-s + (0.794 − 0.606i)21-s − 0.185·23-s + 0.424·25-s + (−0.934 − 0.357i)27-s + (−0.250 + 0.434i)29-s + (0.110 − 0.191i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.913 + 0.406i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.913 + 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0211674 - 0.0995500i\)
\(L(\frac12)\) \(\approx\) \(0.0211674 - 0.0995500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.677 - 1.59i)T \)
7 \( 1 + (0.654 + 2.56i)T \)
good5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.57 + 6.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.01 - 6.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.887T + 23T^{2} \)
29 \( 1 + (1.35 - 2.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.614 + 1.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.26 + 9.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.43 + 2.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.40 + 5.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.38 + 4.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.79 - 8.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.74 - 8.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.49 - 9.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.62T + 71T^{2} \)
73 \( 1 + (-2.01 - 3.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.514 - 0.891i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.26 - 9.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.72 + 2.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.12 - 1.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07252973326532102671548355744, −10.68125875035851265309322784069, −9.720710053333830755537900655693, −8.726773587159799620420930295992, −7.81858808884981183720497822363, −7.22066472834131156422923726414, −5.66940059286169914380110204261, −4.33465972963552211079261394967, −3.95782584088353509653082345579, −2.64788970070140309088380598260, 0.05369844489393475083456499005, 2.24911468217778534456819492911, 3.22319131066144567975883706431, 4.59042707700735706666775137849, 5.96579892943729733285938730926, 6.77489734131823403323469268948, 8.080018405342268656773124925217, 8.214202985338627014625106116840, 9.218089260187309646838031046108, 10.67483938543697708797535555935

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