Properties

Label 2-504-63.20-c1-0-4
Degree $2$
Conductor $504$
Sign $0.0933 - 0.995i$
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  + (−1.31 + 1.13i)3-s + (0.965 + 1.67i)5-s + (2.53 + 0.768i)7-s + (0.441 − 2.96i)9-s + (1.10 + 0.639i)11-s + (2.52 − 1.45i)13-s + (−3.15 − 1.10i)15-s − 0.475·17-s + 6.10i·19-s + (−4.19 + 1.85i)21-s + (−6.51 + 3.75i)23-s + (0.635 − 1.09i)25-s + (2.77 + 4.39i)27-s + (3.76 + 2.17i)29-s + (−4.21 + 2.43i)31-s + ⋯
L(s)  = 1  + (−0.757 + 0.652i)3-s + (0.431 + 0.747i)5-s + (0.956 + 0.290i)7-s + (0.147 − 0.989i)9-s + (0.333 + 0.192i)11-s + (0.701 − 0.404i)13-s + (−0.815 − 0.284i)15-s − 0.115·17-s + 1.40i·19-s + (−0.914 + 0.404i)21-s + (−1.35 + 0.783i)23-s + (0.127 − 0.219i)25-s + (0.534 + 0.845i)27-s + (0.699 + 0.403i)29-s + (−0.757 + 0.437i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0933 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0933 - 0.995i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0933 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972076 + 0.885224i\)
\(L(\frac12)\) \(\approx\) \(0.972076 + 0.885224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.31 - 1.13i)T \)
7 \( 1 + (-2.53 - 0.768i)T \)
good5 \( 1 + (-0.965 - 1.67i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.10 - 0.639i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.52 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.475T + 17T^{2} \)
19 \( 1 - 6.10iT - 19T^{2} \)
23 \( 1 + (6.51 - 3.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.76 - 2.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.21 - 2.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.76T + 37T^{2} \)
41 \( 1 + (1.16 + 2.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 + 8.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.00 - 6.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (1.74 + 3.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.26 + 2.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.602 - 1.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 - 1.84iT - 73T^{2} \)
79 \( 1 + (-8.54 + 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.225 - 0.390i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (9.22 + 5.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.88554512614636792633623425132, −10.48786347904572803833295963536, −9.565890250015097190227636674570, −8.521120013409663992203709631221, −7.46061389108330794670263771821, −6.18202250869995806611431590683, −5.70112029368783243384256151487, −4.48057966269657759307861970240, −3.41557342729288496748834119852, −1.68804909628364085202775733048, 0.964318627427661784403862722871, 2.12244517447906681704817140339, 4.26541265856413050597313131511, 5.06743548582395917073104777498, 6.06581133935962628287228313783, 6.93722109377411980865029214755, 8.066722822715483834928728422635, 8.749293696139438271211621992268, 9.890005325980090744973441748040, 11.06391495601796982606324964944

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