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Properties

Label 2-504-56.53-c1-0-22
Degree $2$
Conductor $504$
Sign $0.959 - 0.281i$
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  + (−0.804 + 1.16i)2-s + (−0.706 − 1.87i)4-s + (2.80 − 1.61i)5-s + (1.47 + 2.19i)7-s + (2.74 + 0.682i)8-s + (−0.371 + 4.56i)10-s + (−2.08 − 1.20i)11-s − 3.09i·13-s + (−3.74 − 0.0456i)14-s + (−3.00 + 2.64i)16-s + (1.97 − 3.42i)17-s + (2.33 − 1.35i)19-s + (−5.01 − 4.10i)20-s + (3.08 − 1.46i)22-s + (1.37 + 2.37i)23-s + ⋯
L(s)  = 1  + (−0.568 + 0.822i)2-s + (−0.353 − 0.935i)4-s + (1.25 − 0.724i)5-s + (0.558 + 0.829i)7-s + (0.970 + 0.241i)8-s + (−0.117 + 1.44i)10-s + (−0.629 − 0.363i)11-s − 0.858i·13-s + (−0.999 − 0.0122i)14-s + (−0.750 + 0.661i)16-s + (0.479 − 0.830i)17-s + (0.536 − 0.309i)19-s + (−1.12 − 0.917i)20-s + (0.657 − 0.311i)22-s + (0.286 + 0.495i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.959 - 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33307 + 0.191617i\)
\(L(\frac12)\) \(\approx\) \(1.33307 + 0.191617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.804 - 1.16i)T \)
3 \( 1 \)
7 \( 1 + (-1.47 - 2.19i)T \)
good5 \( 1 + (-2.80 + 1.61i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.08 + 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + (-1.97 + 3.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.33 + 1.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.37 - 2.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (-1.10 + 1.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.30 + 2.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 + (3.31 + 5.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.23 + 1.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.6 - 6.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.54 - 4.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.01 - 2.89i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 + (-4.77 + 8.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.838 + 1.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 + (-6.98 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
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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

−10.69035603623673547890144391593, −9.701368317598341834929185010578, −9.217166028967177038464190932403, −8.295468834934789072641249531780, −7.53776032344868970146804980250, −6.12490411187292545364130022217, −5.41151313975107907339442479156, −4.96012856957220992483892965721, −2.61750835825301870320645072595, −1.16919402483769843206285757080, 1.50752135460875631962381851789, 2.51912302294638971679461760191, 3.86084716016702536201818709191, 5.06320089851264087121832587194, 6.44518483570827249886735691521, 7.38407317198412694676848717434, 8.280425276273031131472613647029, 9.455284600553046010571637120787, 10.14055117292466942596628004305, 10.64358571515331230691092654475

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