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Properties

Label 2-504-168.5-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.342 - 0.939i$
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  + (−1.11 + 0.875i)2-s + (0.467 − 1.94i)4-s + (3.18 + 1.84i)5-s + (−0.998 + 2.45i)7-s + (1.18 + 2.56i)8-s + (−5.15 + 0.746i)10-s + (0.568 + 0.984i)11-s − 3.62·13-s + (−1.03 − 3.59i)14-s + (−3.56 − 1.81i)16-s + (2.84 + 4.93i)17-s + (2.63 − 4.56i)19-s + (5.06 − 5.33i)20-s + (−1.49 − 0.596i)22-s + (−3.19 − 1.84i)23-s + ⋯
L(s)  = 1  + (−0.785 + 0.619i)2-s + (0.233 − 0.972i)4-s + (1.42 + 0.822i)5-s + (−0.377 + 0.926i)7-s + (0.418 + 0.908i)8-s + (−1.62 + 0.235i)10-s + (0.171 + 0.296i)11-s − 1.00·13-s + (−0.276 − 0.960i)14-s + (−0.890 − 0.454i)16-s + (0.691 + 1.19i)17-s + (0.604 − 1.04i)19-s + (1.13 − 1.19i)20-s + (−0.318 − 0.127i)22-s + (−0.665 − 0.384i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641609 + 0.916883i\)
\(L(\frac12)\) \(\approx\) \(0.641609 + 0.916883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.875i)T \)
3 \( 1 \)
7 \( 1 + (0.998 - 2.45i)T \)
good5 \( 1 + (-3.18 - 1.84i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.568 - 0.984i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.62T + 13T^{2} \)
17 \( 1 + (-2.84 - 4.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.63 + 4.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.19 + 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + (5.52 - 3.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.63 - 4.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 7.00iT - 43T^{2} \)
47 \( 1 + (3.98 - 6.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.84 - 4.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.813 - 0.469i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.98 + 3.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.18 - 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 + (-6.72 + 3.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.68 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.89iT - 83T^{2} \)
89 \( 1 + (-1.04 + 1.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.79331012916772411915891820746, −9.967000542829315463804088151007, −9.572255696535613568691487174167, −8.675065826694528397752897631001, −7.46785669033518787870249498164, −6.53732158993576346060321600400, −5.91648343781809836590380504977, −5.04717668371932417002539043974, −2.81289510304429161287659066094, −1.86614768440356046213365076254, 0.882809712193354819918681545638, 2.16091386418057751186690454412, 3.53563549346616491199194904351, 4.91004290411805056586002609664, 6.03513056257930263215157305703, 7.24680461041643925066803227745, 8.035866296898297473740629821081, 9.378971696932757870489740635486, 9.681221028422205932913435393353, 10.25743726437480353140188322764

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