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Properties

Label 2-504-8.5-c1-0-5
Degree $2$
Conductor $504$
Sign $0.947 - 0.320i$
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  + (−0.835 − 1.14i)2-s + (−0.602 + 1.90i)4-s − 0.467i·5-s + 7-s + (2.67 − 0.907i)8-s + (−0.532 + 0.390i)10-s + 4.87i·11-s + 4.56i·13-s + (−0.835 − 1.14i)14-s + (−3.27 − 2.29i)16-s − 6.09·17-s − 1.34i·19-s + (0.890 + 0.281i)20-s + (5.56 − 4.07i)22-s + 4.09·23-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.301 + 0.953i)4-s − 0.208i·5-s + 0.377·7-s + (0.947 − 0.320i)8-s + (−0.168 + 0.123i)10-s + 1.47i·11-s + 1.26i·13-s + (−0.223 − 0.304i)14-s + (−0.818 − 0.574i)16-s − 1.47·17-s − 0.308i·19-s + (0.199 + 0.0629i)20-s + (1.18 − 0.869i)22-s + 0.854·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.947 - 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946389 + 0.155879i\)
\(L(\frac12)\) \(\approx\) \(0.946389 + 0.155879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.835 + 1.14i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 0.467iT - 5T^{2} \)
11 \( 1 - 4.87iT - 11T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
17 \( 1 + 6.09T + 17T^{2} \)
19 \( 1 + 1.34iT - 19T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 - 7.78iT - 29T^{2} \)
31 \( 1 - 4.40T + 31T^{2} \)
37 \( 1 - 4.40iT - 37T^{2} \)
41 \( 1 - 6.09T + 41T^{2} \)
43 \( 1 + 4.15iT - 43T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 + 1.34iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 5.49iT - 61T^{2} \)
67 \( 1 + 5.90iT - 67T^{2} \)
71 \( 1 - 4.72T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−10.96647514679362197959886543698, −10.13246663306871724449565007812, −9.015580650195072121931139602712, −8.798687918868229208108943470025, −7.29916014184656595570276339248, −6.81473296267338362779378086119, −4.78080564686460801494978131890, −4.32453492078120073288991482209, −2.63298668947660401063181106926, −1.54322462237772414681169712156, 0.75310448963410233146299673247, 2.72351971551395425755218601783, 4.35997836515755173218434722486, 5.54851865334901671896548901902, 6.25562340471470332302943198835, 7.32673088953046774691157880182, 8.297391722112231020441561851455, 8.777505045588127815671695920931, 9.905963152468702153660583586449, 10.95928915345836385092241797604

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