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Properties

Label 2-504-63.47-c1-0-11
Degree $2$
Conductor $504$
Sign $0.682 - 0.731i$
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  + (1.24 + 1.20i)3-s + 2.04·5-s + (1.41 + 2.23i)7-s + (0.102 + 2.99i)9-s − 5.90i·11-s + (−0.139 + 0.0804i)13-s + (2.55 + 2.46i)15-s + (2.77 + 4.81i)17-s + (−4.02 − 2.32i)19-s + (−0.932 + 4.48i)21-s − 0.433i·23-s − 0.801·25-s + (−3.48 + 3.85i)27-s + (−1.95 − 1.12i)29-s + (−2.57 − 1.48i)31-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)3-s + 0.916·5-s + (0.534 + 0.845i)7-s + (0.0341 + 0.999i)9-s − 1.78i·11-s + (−0.0386 + 0.0223i)13-s + (0.658 + 0.636i)15-s + (0.674 + 1.16i)17-s + (−0.922 − 0.532i)19-s + (−0.203 + 0.979i)21-s − 0.0903i·23-s − 0.160·25-s + (−0.669 + 0.742i)27-s + (−0.363 − 0.209i)29-s + (−0.463 − 0.267i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96169 + 0.852496i\)
\(L(\frac12)\) \(\approx\) \(1.96169 + 0.852496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 1.20i)T \)
7 \( 1 + (-1.41 - 2.23i)T \)
good5 \( 1 - 2.04T + 5T^{2} \)
11 \( 1 + 5.90iT - 11T^{2} \)
13 \( 1 + (0.139 - 0.0804i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.77 - 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.02 + 2.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.433iT - 23T^{2} \)
29 \( 1 + (1.95 + 1.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.57 + 1.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.17 + 3.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.35 - 4.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.82 + 3.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0650 - 0.112i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.7 + 6.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.22 - 5.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.98 - 3.45i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.64 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.48iT - 71T^{2} \)
73 \( 1 + (2.60 - 1.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.69 + 15.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.62 + 13.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.04 + 7.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.61 - 1.50i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.83380219995794941994696222744, −10.17576695297354795506745307458, −8.991352566336849918357282628856, −8.692984506195937729434295323466, −7.74543936457571236950787217524, −5.96320872549319632422489221057, −5.63816954097612012214516667631, −4.22519617631129815922253571096, −3.01348929689538562267884301172, −1.92631491442266358942918948780, 1.48047082904226530229898227683, 2.40653372910609722132481215029, 3.96786109687723200036510795854, 5.09504600990464021784076073076, 6.41931348071990498776766554565, 7.32620350146578919213699105913, 7.84270334855691085722027762261, 9.182339413071715302780625452595, 9.782925515312440841749404505252, 10.58685345588765892326408961879

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