Properties

Label 2-504-56.53-c1-0-8
Degree $2$
Conductor $504$
Sign $0.960 - 0.279i$
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  + (0.264 − 1.38i)2-s + (−1.85 − 0.735i)4-s + (−0.937 + 0.541i)5-s + (−1.62 + 2.09i)7-s + (−1.51 + 2.38i)8-s + (0.503 + 1.44i)10-s + (4.52 + 2.61i)11-s + 4.43i·13-s + (2.47 + 2.80i)14-s + (2.91 + 2.73i)16-s + (3.65 − 6.33i)17-s + (−2.42 + 1.39i)19-s + (2.14 − 0.317i)20-s + (4.82 − 5.59i)22-s + (1.51 + 2.62i)23-s + ⋯
L(s)  = 1  + (0.187 − 0.982i)2-s + (−0.929 − 0.367i)4-s + (−0.419 + 0.242i)5-s + (−0.612 + 0.790i)7-s + (−0.535 + 0.844i)8-s + (0.159 + 0.457i)10-s + (1.36 + 0.787i)11-s + 1.23i·13-s + (0.661 + 0.749i)14-s + (0.729 + 0.684i)16-s + (0.886 − 1.53i)17-s + (−0.555 + 0.320i)19-s + (0.478 − 0.0708i)20-s + (1.02 − 1.19i)22-s + (0.315 + 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.960 - 0.279i$
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: \((2,\ 504,\ (\ :1/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10625 + 0.157523i\)
\(L(\frac12)\) \(\approx\) \(1.10625 + 0.157523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.264 + 1.38i)T \)
3 \( 1 \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (0.937 - 0.541i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.52 - 2.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.43iT - 13T^{2} \)
17 \( 1 + (-3.65 + 6.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 - 1.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.51 - 2.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 + (-3.20 + 5.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.85 - 4.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 + (-4.54 - 7.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.71 + 0.989i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.01 - 3.47i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.85 + 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.26 - 3.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.53T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.621 - 1.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (3.02 + 5.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.82T + 97T^{2} \)
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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

−11.36021966325880614294877957885, −9.818490685244111906910365523095, −9.507976678866113591626040912726, −8.673754006578163579888297512893, −7.25140022237365598856591887071, −6.29555317727994288776161975954, −5.03507452697593015981679105877, −3.99310962438167581222739020296, −3.00744138143748393935279619943, −1.63835002451740464465185633333, 0.68366179958327286120407137546, 3.53200163423502436862215804751, 3.98123564871043195555015846411, 5.42994575708956035386851758729, 6.35955509322366599869086647499, 7.08431181087409135883018760984, 8.320666376318774920958025670401, 8.627050577559442980692141337854, 9.998505825486356044514025377188, 10.62283888285652678268056389912

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