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Properties

Label 2-504-63.25-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.580 - 0.814i$
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  + 1.73i·3-s + (−0.5 + 0.866i)5-s + (2 + 1.73i)7-s − 2.99·9-s + (1.5 + 2.59i)11-s + (−0.5 − 0.866i)13-s + (−1.49 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (−2.5 − 4.33i)19-s + (−2.99 + 3.46i)21-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s − 5.19i·27-s + (−4.5 + 7.79i)29-s + 4·31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.223 + 0.387i)5-s + (0.755 + 0.654i)7-s − 0.999·9-s + (0.452 + 0.783i)11-s + (−0.138 − 0.240i)13-s + (−0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (−0.573 − 0.993i)19-s + (−0.654 + 0.755i)21-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s − 0.999i·27-s + (−0.835 + 1.44i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586961 + 1.13905i\)
\(L(\frac12)\) \(\approx\) \(0.586961 + 1.13905i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-6.5 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96872780422836244636053302551, −10.59752003967872385689095045267, −9.275099372735580977504924546695, −8.853805338053689362431630598784, −7.71437371029749242831576087957, −6.60613027272634347985098878976, −5.37327248727214421913125792438, −4.60987307763126870548943451470, −3.49965207096108577145407390252, −2.15969662547487680131226760090, 0.78188985092565427348299079964, 2.15456553030658103953263210985, 3.76837441936176330092345489323, 4.91996073971280003327592379027, 6.14708224125194946004697846480, 6.95035636689760271002130351288, 8.138275876811640239575296894788, 8.355025850212131264009730092336, 9.668425574889946498384454111469, 10.87582778752665725164771423053

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