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Properties

Label 2-504-63.4-c1-0-2
Degree $2$
Conductor $504$
Sign $0.615 - 0.788i$
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  + (−0.849 − 1.50i)3-s − 1.58·5-s + (−1.80 + 1.93i)7-s + (−1.55 + 2.56i)9-s + 5.17·11-s + (−0.681 + 1.18i)13-s + (1.34 + 2.38i)15-s + (−2.30 + 3.99i)17-s + (0.0321 + 0.0557i)19-s + (4.45 + 1.08i)21-s + 6.74·23-s − 2.49·25-s + (5.19 + 0.166i)27-s + (4.70 + 8.15i)29-s + (1.33 + 2.30i)31-s + ⋯
L(s)  = 1  + (−0.490 − 0.871i)3-s − 0.707·5-s + (−0.683 + 0.729i)7-s + (−0.518 + 0.855i)9-s + 1.55·11-s + (−0.189 + 0.327i)13-s + (0.347 + 0.616i)15-s + (−0.559 + 0.969i)17-s + (0.00738 + 0.0127i)19-s + (0.971 + 0.237i)21-s + 1.40·23-s − 0.499·25-s + (0.999 + 0.0320i)27-s + (0.874 + 1.51i)29-s + (0.239 + 0.414i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733959 + 0.357997i\)
\(L(\frac12)\) \(\approx\) \(0.733959 + 0.357997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.849 + 1.50i)T \)
7 \( 1 + (1.80 - 1.93i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
13 \( 1 + (0.681 - 1.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.30 - 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0321 - 0.0557i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 + (-4.70 - 8.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.33 - 2.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.858 - 1.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.12 + 8.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.60 - 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.479 - 0.831i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.66 - 8.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.19 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.24 - 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.26 - 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.08 + 8.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.12 + 7.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.26 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.38192137025946072762885685337, −10.29943398375088834731277474253, −8.944130019025096627744392079529, −8.531090515041616292902378681605, −7.06663776422068291615322971520, −6.66718284139504054001632182899, −5.64861427986854352048461803932, −4.33551520391233702449658011331, −3.04062394653424015772146430382, −1.44892528034718364303863730861, 0.56512606423638210165611545500, 3.16942768427714371386764826414, 4.05707480017478402401439059570, 4.83469434201374006061754449397, 6.31243749467952376073227507176, 6.90549968483241425623329616303, 8.156095512811796221045969253780, 9.409293127231843209476268289330, 9.707333867463478204937192525423, 10.94038785014325519540222379330

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