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Properties

Label 2-504-168.11-c1-0-29
Degree $2$
Conductor $504$
Sign $-0.932 - 0.360i$
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  + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (−0.914 − 1.58i)5-s + (0.358 − 2.62i)7-s + (−2 + 1.99i)8-s + (−2.49 + 0.669i)10-s + (−1.37 − 0.792i)11-s + 2.58i·13-s + (−3.44 − 1.44i)14-s + (1.99 + 3.46i)16-s + (−5.40 − 3.12i)17-s + (1.29 + 2.23i)19-s + 3.65i·20-s + (−1.58 + 1.58i)22-s + (−0.292 − 0.507i)23-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.408 − 0.708i)5-s + (0.135 − 0.990i)7-s + (−0.707 + 0.707i)8-s + (−0.789 + 0.211i)10-s + (−0.414 − 0.239i)11-s + 0.717i·13-s + (−0.921 − 0.387i)14-s + (0.499 + 0.866i)16-s + (−1.31 − 0.757i)17-s + (0.296 + 0.513i)19-s + 0.817i·20-s + (−0.338 + 0.338i)22-s + (−0.0610 − 0.105i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163410 + 0.874905i\)
\(L(\frac12)\) \(\approx\) \(0.163410 + 0.874905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 \)
7 \( 1 + (-0.358 + 2.62i)T \)
good5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.37 + 0.792i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.58iT - 13T^{2} \)
17 \( 1 + (5.40 + 3.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.29 - 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.292 + 0.507i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (3.82 + 2.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.85 - 4.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (-1.29 - 2.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.08 + 5.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.98 + 4.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.40 + 3.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.82 + 6.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (-3.65 + 6.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.98 + 4.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.07iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.71497735802751838532126332549, −9.588989155480588167625431831242, −8.864858570219808817954051796739, −7.902721412467828002157454259020, −6.74463502701365979947675675060, −5.29260741726497788192590266025, −4.44199618582839479424269114172, −3.65099646496590852516713332112, −2.05428228623680504142667274559, −0.48083094987114952945340903061, 2.59191849160369067858048674057, 3.78130237213288956828436541627, 5.04487998418538821154383339872, 5.87255133648010252016670209339, 6.89265426537594771466513565623, 7.66810541339001513877543849907, 8.617841145746596944552847753678, 9.313725518102850206338131216515, 10.59478638596050278087033104514, 11.39279492177534093581107919322

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