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Properties

Label 2-504-168.107-c1-0-19
Degree $2$
Conductor $504$
Sign $0.910 + 0.413i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (1.91 − 3.31i)5-s + (2.09 + 1.62i)7-s + (−1.99 + 2i)8-s + (−1.40 + 5.22i)10-s + (3.82 − 2.20i)11-s + 5.41i·13-s + (−3.44 − 1.44i)14-s + (1.99 − 3.46i)16-s + (−1.94 + 1.12i)17-s + (2.70 − 4.68i)19-s − 7.65i·20-s + (−4.41 + 4.41i)22-s + (−1.70 + 2.95i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.856 − 1.48i)5-s + (0.790 + 0.612i)7-s + (−0.707 + 0.707i)8-s + (−0.443 + 1.65i)10-s + (1.15 − 0.665i)11-s + 1.50i·13-s + (−0.921 − 0.387i)14-s + (0.499 − 0.866i)16-s + (−0.471 + 0.271i)17-s + (0.621 − 1.07i)19-s − 1.71i·20-s + (−0.941 + 0.941i)22-s + (−0.355 + 0.616i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20678 - 0.261197i\)
\(L(\frac12)\) \(\approx\) \(1.20678 - 0.261197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
7 \( 1 + (-2.09 - 1.62i)T \)
good5 \( 1 + (-1.91 + 3.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.82 + 2.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.41iT - 13T^{2} \)
17 \( 1 + (1.94 - 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.70 + 4.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.70 - 2.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-1.37 + 0.792i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.39 + 2.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 - 8.58T + 43T^{2} \)
47 \( 1 + (-2.70 + 4.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.91 - 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.25 - 3.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.94 - 1.12i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.82 + 3.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.89T + 71T^{2} \)
73 \( 1 + (7.65 + 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.25 - 3.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.07iT - 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−10.83811289839376699627033721153, −9.372732798127262258686434980508, −9.083118754869868632689732600957, −8.652989294202792004515988616792, −7.35963472156856375978373108273, −6.18754412807856396319350646781, −5.46093195703315390123677283590, −4.35258915740547555866315320455, −2.10519292313465868394349218341, −1.23766571384124056219275229533, 1.50028654765216653400101629310, 2.69596376451382903251042844496, 3.85323154895247906046203520427, 5.69311137709621297593275950435, 6.69216007831351849850863218162, 7.36962495032617266388504809346, 8.224704985651271157191617616577, 9.513566902245808670558108765419, 10.17740939507246928695054436366, 10.70862121809848720831327170825

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