Properties

Label 2-504-63.38-c1-0-14
Degree $2$
Conductor $504$
Sign $0.810 - 0.585i$
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  + (0.859 + 1.50i)3-s + (1.29 − 2.24i)5-s + (1.66 + 2.05i)7-s + (−1.52 + 2.58i)9-s + (3.06 − 1.77i)11-s + (−2.74 + 1.58i)13-s + (4.49 + 0.0189i)15-s + (0.487 − 0.844i)17-s + (2.11 − 1.21i)19-s + (−1.65 + 4.27i)21-s + (2.92 + 1.68i)23-s + (−0.862 − 1.49i)25-s + (−5.19 − 0.0658i)27-s + (0.267 + 0.154i)29-s − 5.03i·31-s + ⋯
L(s)  = 1  + (0.496 + 0.868i)3-s + (0.579 − 1.00i)5-s + (0.630 + 0.776i)7-s + (−0.507 + 0.861i)9-s + (0.925 − 0.534i)11-s + (−0.760 + 0.438i)13-s + (1.15 + 0.00490i)15-s + (0.118 − 0.204i)17-s + (0.484 − 0.279i)19-s + (−0.360 + 0.932i)21-s + (0.609 + 0.352i)23-s + (−0.172 − 0.298i)25-s + (−0.999 − 0.0126i)27-s + (0.0497 + 0.0286i)29-s − 0.904i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86421 + 0.603254i\)
\(L(\frac12)\) \(\approx\) \(1.86421 + 0.603254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.859 - 1.50i)T \)
7 \( 1 + (-1.66 - 2.05i)T \)
good5 \( 1 + (-1.29 + 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.06 + 1.77i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.74 - 1.58i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.487 + 0.844i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.11 + 1.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.92 - 1.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.267 - 0.154i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.03iT - 31T^{2} \)
37 \( 1 + (3.47 + 6.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.08 - 10.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.47 - 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.86T + 47T^{2} \)
53 \( 1 + (7.81 + 4.51i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.439T + 59T^{2} \)
61 \( 1 + 3.94iT - 61T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 + 5.25iT - 71T^{2} \)
73 \( 1 + (14.0 + 8.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.99T + 79T^{2} \)
83 \( 1 + (-7.23 + 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.31 + 4.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.4 + 7.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.15178493486289027128998853543, −9.626897658890449012781737944993, −9.363457746362626455583958763249, −8.618672217795628754578284760979, −7.69116470902714237409725807712, −6.10631433326521784695246693166, −5.11800929752180190583641742703, −4.52965786231952226100310755057, −3.05748035399527349201351063116, −1.67899008025513598654091151587, 1.41557443264600942954356119025, 2.62321636788855632757208368206, 3.79436528087988232891702205471, 5.28485042239972966807565113478, 6.65912775741975774919981073204, 7.04113328063023074862676768237, 7.938619595234278414317673468579, 9.015557886734449876310320717332, 10.04023690254872701333377397304, 10.71802642287011085442229059521

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