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Properties

Label 2-504-168.107-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.0144 + 0.999i$
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  + (−0.576 + 1.29i)2-s + (−1.33 − 1.48i)4-s + (−1.88 + 3.26i)5-s + (−2.23 + 1.41i)7-s + (2.69 − 0.868i)8-s + (−3.12 − 4.30i)10-s + (−1.47 + 0.849i)11-s − 5.64i·13-s + (−0.543 − 3.70i)14-s + (−0.428 + 3.97i)16-s + (2.26 − 1.30i)17-s + (−1.18 + 2.04i)19-s + (7.36 − 1.55i)20-s + (−0.249 − 2.38i)22-s + (0.653 − 1.13i)23-s + ⋯
L(s)  = 1  + (−0.407 + 0.913i)2-s + (−0.668 − 0.743i)4-s + (−0.841 + 1.45i)5-s + (−0.844 + 0.535i)7-s + (0.951 − 0.307i)8-s + (−0.988 − 1.36i)10-s + (−0.443 + 0.256i)11-s − 1.56i·13-s + (−0.145 − 0.989i)14-s + (−0.107 + 0.994i)16-s + (0.548 − 0.316i)17-s + (−0.270 + 0.469i)19-s + (1.64 − 0.347i)20-s + (−0.0532 − 0.509i)22-s + (0.136 − 0.236i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0144 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0144 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0144 + 0.999i$
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: \((2,\ 504,\ (\ :1/2),\ -0.0144 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0368580 - 0.0373950i\)
\(L(\frac12)\) \(\approx\) \(0.0368580 - 0.0373950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.576 - 1.29i)T \)
3 \( 1 \)
7 \( 1 + (2.23 - 1.41i)T \)
good5 \( 1 + (1.88 - 3.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.47 - 0.849i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 + (-2.26 + 1.30i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.18 - 2.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.653 + 1.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 + (-4.75 + 2.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.20 + 3.00i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.10iT - 41T^{2} \)
43 \( 1 + 6.49T + 43T^{2} \)
47 \( 1 + (3.53 - 6.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.488 - 0.846i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.67 - 3.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.64 - 4.41i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.69 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + (1.09 + 1.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (15.0 + 8.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (9.58 + 5.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−11.36439162640275962240750102375, −10.29126359257331752098325200003, −10.06234779119935192978860185189, −8.693978001662748631682760551449, −7.73513050041361537646100630802, −7.21464839537244891222036989320, −6.19637285408225071492583184926, −5.40769695430693246785423259791, −3.78037949478506805039647197390, −2.77623398390321227243140632143, 0.03677894977012555597812942307, 1.51977525528321364069748724075, 3.36642411929662424494880252184, 4.22787682566681030208010876974, 5.08249036966115687027909597894, 6.79566827566409969345605959377, 7.87453710463878962870418133863, 8.666771502084110205372367161769, 9.358905468292328903954819839062, 10.16070954058907436413755292680

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