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Properties

Label 2-504-56.53-c1-0-36
Degree $2$
Conductor $504$
Sign $-0.642 - 0.766i$
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  + (0.453 − 1.33i)2-s + (−1.58 − 1.21i)4-s + (−0.476 + 0.274i)5-s + (−2.60 + 0.447i)7-s + (−2.34 + 1.57i)8-s + (0.152 + 0.762i)10-s + (−2.07 − 1.19i)11-s − 3.96i·13-s + (−0.583 + 3.69i)14-s + (1.04 + 3.86i)16-s + (−2.10 + 3.65i)17-s + (−5.75 + 3.32i)19-s + (1.09 + 0.142i)20-s + (−2.54 + 2.23i)22-s + (−1.17 − 2.03i)23-s + ⋯
L(s)  = 1  + (0.320 − 0.947i)2-s + (−0.794 − 0.607i)4-s + (−0.212 + 0.122i)5-s + (−0.985 + 0.169i)7-s + (−0.830 + 0.557i)8-s + (0.0481 + 0.241i)10-s + (−0.624 − 0.360i)11-s − 1.10i·13-s + (−0.155 + 0.987i)14-s + (0.261 + 0.965i)16-s + (−0.511 + 0.885i)17-s + (−1.31 + 0.761i)19-s + (0.243 + 0.0317i)20-s + (−0.541 + 0.475i)22-s + (−0.244 − 0.424i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 504,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105845 + 0.226839i\)
\(L(\frac12)\) \(\approx\) \(0.105845 + 0.226839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 1.33i)T \)
3 \( 1 \)
7 \( 1 + (2.60 - 0.447i)T \)
good5 \( 1 + (0.476 - 0.274i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 + 1.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.96iT - 13T^{2} \)
17 \( 1 + (2.10 - 3.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.75 - 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 + 2.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.21iT - 29T^{2} \)
31 \( 1 + (0.433 - 0.750i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.229 + 0.132i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 5.35iT - 43T^{2} \)
47 \( 1 + (-1.29 - 2.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.36 + 5.40i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.26 + 1.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.18 + 3.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.31 + 1.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (2.33 - 4.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.308 + 0.533i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.09iT - 83T^{2} \)
89 \( 1 + (3.19 + 5.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.44818919365861499122149782463, −9.791511652812719712946426553345, −8.677603768294454813552355931839, −7.893844363499898809435878601237, −6.26269342770934123418753058631, −5.68304487087781118808131741627, −4.23896465232255186157576881913, −3.33632743894007879018757611794, −2.23286352691116662122834331832, −0.12589070121495988920342771513, 2.70018860594269314253824647333, 4.08020643850402007561098113545, 4.83728998073021994736477782010, 6.16196877468697610269631085895, 6.85818047952258138131522261054, 7.64304546108550304429443564053, 8.868844081372416885247401981109, 9.368429096899381065127712809939, 10.48017776065111462079570200301, 11.66049845953942140334847537987

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