Properties

Label 2-504-56.27-c1-0-18
Degree $2$
Conductor $504$
Sign $0.965 + 0.261i$
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  + (0.185 + 1.40i)2-s + (−1.93 + 0.520i)4-s − 3.84·5-s + (1.62 − 2.09i)7-s + (−1.08 − 2.61i)8-s + (−0.713 − 5.38i)10-s + 4.54·11-s − 1.81·13-s + (3.23 + 1.88i)14-s + (3.45 − 2.00i)16-s − 3.49i·17-s − 1.68i·19-s + (7.42 − 2.00i)20-s + (0.843 + 6.37i)22-s − 5.00i·23-s + ⋯
L(s)  = 1  + (0.131 + 0.991i)2-s + (−0.965 + 0.260i)4-s − 1.71·5-s + (0.612 − 0.790i)7-s + (−0.384 − 0.923i)8-s + (−0.225 − 1.70i)10-s + 1.37·11-s − 0.503·13-s + (0.863 + 0.503i)14-s + (0.864 − 0.502i)16-s − 0.846i·17-s − 0.387i·19-s + (1.66 − 0.447i)20-s + (0.179 + 1.35i)22-s − 1.04i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894753 - 0.119219i\)
\(L(\frac12)\) \(\approx\) \(0.894753 - 0.119219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.185 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-1.62 + 2.09i)T \)
good5 \( 1 + 3.84T + 5T^{2} \)
11 \( 1 - 4.54T + 11T^{2} \)
13 \( 1 + 1.81T + 13T^{2} \)
17 \( 1 + 3.49iT - 17T^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + 5.00iT - 23T^{2} \)
29 \( 1 - 1.81iT - 29T^{2} \)
31 \( 1 - 5.34T + 31T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 + 8.97iT - 41T^{2} \)
43 \( 1 + 8.03T + 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
53 \( 1 + 5.87iT - 53T^{2} \)
59 \( 1 + 8.46iT - 59T^{2} \)
61 \( 1 - 3.01T + 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + 1.47iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 + 2.97iT - 79T^{2} \)
83 \( 1 - 10.5iT - 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−11.06959129880822465276811180782, −9.819653185350928698315933187681, −8.684079395490538105664496963574, −8.094468812433165862873356803539, −7.10494945409329003770241706896, −6.75899166253698609207329313567, −4.94819816060509044362698813190, −4.31399113361726651957230967265, −3.46905522007591780911810671884, −0.59086269258208608820941345088, 1.46290277102197709592504359224, 3.14111614218254019802189894246, 4.07214928643961585187356259841, 4.83363679951657804090075617401, 6.22244570391061941207772232914, 7.69512144984424680116299174631, 8.379193777971806736666162579572, 9.152451352016320067179002556489, 10.23556487282386687162599258965, 11.42372466384549480234579342162

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