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Properties

Label 2-504-63.47-c1-0-12
Degree $2$
Conductor $504$
Sign $0.962 + 0.269i$
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  + (1.64 − 0.550i)3-s − 1.28·5-s + (1.10 + 2.40i)7-s + (2.39 − 1.80i)9-s − 3.61i·11-s + (3.48 − 2.01i)13-s + (−2.11 + 0.708i)15-s + (0.828 + 1.43i)17-s + (5.15 + 2.97i)19-s + (3.13 + 3.34i)21-s − 0.429i·23-s − 3.34·25-s + (2.93 − 4.28i)27-s + (6.39 + 3.69i)29-s + (−0.841 − 0.485i)31-s + ⋯
L(s)  = 1  + (0.948 − 0.317i)3-s − 0.575·5-s + (0.415 + 0.909i)7-s + (0.798 − 0.602i)9-s − 1.09i·11-s + (0.967 − 0.558i)13-s + (−0.546 + 0.182i)15-s + (0.200 + 0.347i)17-s + (1.18 + 0.682i)19-s + (0.683 + 0.730i)21-s − 0.0896i·23-s − 0.668·25-s + (0.565 − 0.824i)27-s + (1.18 + 0.685i)29-s + (−0.151 − 0.0872i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93975 - 0.266325i\)
\(L(\frac12)\) \(\approx\) \(1.93975 - 0.266325i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.550i)T \)
7 \( 1 + (-1.10 - 2.40i)T \)
good5 \( 1 + 1.28T + 5T^{2} \)
11 \( 1 + 3.61iT - 11T^{2} \)
13 \( 1 + (-3.48 + 2.01i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.828 - 1.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.15 - 2.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.429iT - 23T^{2} \)
29 \( 1 + (-6.39 - 3.69i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.841 + 0.485i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.16 - 8.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.15 + 8.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.67 + 6.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.01 + 6.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.4 - 6.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.618 - 1.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.75 - 3.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.10 + 1.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.66iT - 71T^{2} \)
73 \( 1 + (1.67 - 0.965i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.26 - 3.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.701 + 1.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.81 - 8.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.20 + 4.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.91738552972551822659182823844, −9.905395432550753858366548685011, −8.599368953942390373156859176031, −8.472956812273931942158397681296, −7.53782517213384065039354672111, −6.27610005313556221458373459105, −5.29837543940216300882090419780, −3.71404225679866058361637440608, −3.03372779211318631680335545851, −1.41515214812111150890848297194, 1.55068558934611127358402889863, 3.15760395243783630170842201973, 4.18036361130322583382066135794, 4.86013153777831854839897905152, 6.67563564925632499635105575706, 7.57836253910325504243086145105, 8.088371609749700394477625079313, 9.296933658714234765438661818161, 9.890514375166629607404166279385, 10.94242102943724006778682059476

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