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Properties

Label 2-504-9.4-c1-0-8
Degree $2$
Conductor $504$
Sign $0.992 - 0.124i$
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  + (−1.25 + 1.19i)3-s + (0.846 − 1.46i)5-s + (0.5 + 0.866i)7-s + (0.150 − 2.99i)9-s + (0.474 + 0.822i)11-s + (1.69 − 2.93i)13-s + (0.687 + 2.84i)15-s − 0.300·17-s + 5.73·19-s + (−1.66 − 0.490i)21-s + (1.17 − 2.03i)23-s + (1.06 + 1.84i)25-s + (3.38 + 3.93i)27-s + (1.01 + 1.76i)29-s + (−0.522 + 0.905i)31-s + ⋯
L(s)  = 1  + (−0.724 + 0.689i)3-s + (0.378 − 0.655i)5-s + (0.188 + 0.327i)7-s + (0.0501 − 0.998i)9-s + (0.143 + 0.247i)11-s + (0.470 − 0.814i)13-s + (0.177 + 0.735i)15-s − 0.0729·17-s + 1.31·19-s + (−0.362 − 0.106i)21-s + (0.244 − 0.423i)23-s + (0.213 + 0.369i)25-s + (0.651 + 0.758i)27-s + (0.189 + 0.327i)29-s + (−0.0938 + 0.162i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27790 + 0.0795873i\)
\(L(\frac12)\) \(\approx\) \(1.27790 + 0.0795873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 1.19i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.846 + 1.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.474 - 0.822i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.69 + 2.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.300T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 + (-1.17 + 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.01 - 1.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.522 - 0.905i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.22T + 37T^{2} \)
41 \( 1 + (-2.56 + 4.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.25 - 9.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.24 + 5.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.86T + 53T^{2} \)
59 \( 1 + (-4.46 + 7.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.256 + 0.443i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.63 - 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.04T + 71T^{2} \)
73 \( 1 + 1.59T + 73T^{2} \)
79 \( 1 + (6.43 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.98 + 3.43i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.24T + 89T^{2} \)
97 \( 1 + (5.02 + 8.70i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96966935728231343141367653783, −10.00110824401446811618420506996, −9.307499505107241141084263341978, −8.479620057748325065479698849510, −7.22883677843161540482047651144, −5.96354462998806767227984092196, −5.33473907118710790653890705334, −4.43618074622070580946554376186, −3.09839677271246997707148280244, −1.10728347527525088215319670116, 1.23986160624913363632404373819, 2.68688176556172322915324669396, 4.23513545567650018357849503900, 5.50664539372799846990820960655, 6.34496466847412255271708428995, 7.12708609250943956821434350928, 7.949098720501939912292145447882, 9.211161389096922358555321722284, 10.19589809981912776433903978028, 11.10414812193419237279329368890

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