Properties

Label 2-504-56.3-c1-0-19
Degree $2$
Conductor $504$
Sign $0.979 - 0.203i$
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  + (−0.169 + 1.40i)2-s + (−1.94 − 0.477i)4-s + (−0.345 − 0.597i)5-s + (−2.63 − 0.222i)7-s + (0.999 − 2.64i)8-s + (0.897 − 0.382i)10-s + (1.63 − 2.82i)11-s + 5.27·13-s + (0.760 − 3.66i)14-s + (3.54 + 1.85i)16-s + (2.20 + 1.27i)17-s + (−0.484 + 0.279i)19-s + (0.385 + 1.32i)20-s + (3.68 + 2.76i)22-s + (2.50 − 1.44i)23-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)2-s + (−0.971 − 0.238i)4-s + (−0.154 − 0.267i)5-s + (−0.996 − 0.0840i)7-s + (0.353 − 0.935i)8-s + (0.283 − 0.121i)10-s + (0.491 − 0.851i)11-s + 1.46·13-s + (0.203 − 0.979i)14-s + (0.886 + 0.463i)16-s + (0.534 + 0.308i)17-s + (−0.111 + 0.0642i)19-s + (0.0861 + 0.296i)20-s + (0.786 + 0.590i)22-s + (0.522 − 0.301i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.979 - 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08585 + 0.111613i\)
\(L(\frac12)\) \(\approx\) \(1.08585 + 0.111613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.169 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (2.63 + 0.222i)T \)
good5 \( 1 + (0.345 + 0.597i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.27T + 13T^{2} \)
17 \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.484 - 0.279i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.444iT - 29T^{2} \)
31 \( 1 + (-4.45 + 7.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.00 - 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.76iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.20 + 3.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.17 - 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.59 + 4.96i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.45 - 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (5.28 + 3.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.01 - 2.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.42iT - 97T^{2} \)
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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

−10.70486710689050677179090741084, −9.923394781286195586667942034529, −8.737837623639323930895632948967, −8.526878728003689829105662952151, −7.18374772855856730624466088962, −6.26551600524358938421566384694, −5.72104049849678213089545731491, −4.22162790974360749136260610525, −3.37294011858378742114857462209, −0.823192453831704747953172461389, 1.34757179564338085050056674305, 3.01633659155385262486466392717, 3.72573403635759336872658722406, 4.99147494016450177703411566182, 6.28873478521919132403675187311, 7.28414958549030961868116671350, 8.556322387738904818975822301919, 9.276239461043339666424198688162, 10.07863270178289041261243877651, 10.87017822609972326311501948194

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