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Properties

Label 2-504-63.47-c1-0-18
Degree $2$
Conductor $504$
Sign $0.636 + 0.771i$
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  + (0.930 − 1.46i)3-s + 1.36·5-s + (2.64 − 0.0810i)7-s + (−1.26 − 2.71i)9-s + 1.20i·11-s + (0.639 − 0.369i)13-s + (1.27 − 1.99i)15-s + (−0.693 − 1.20i)17-s + (2.81 + 1.62i)19-s + (2.34 − 3.93i)21-s + 3.81i·23-s − 3.12·25-s + (−5.15 − 0.677i)27-s + (−3.50 − 2.02i)29-s + (1.02 + 0.594i)31-s + ⋯
L(s)  = 1  + (0.537 − 0.843i)3-s + 0.611·5-s + (0.999 − 0.0306i)7-s + (−0.422 − 0.906i)9-s + 0.362i·11-s + (0.177 − 0.102i)13-s + (0.328 − 0.516i)15-s + (−0.168 − 0.291i)17-s + (0.646 + 0.373i)19-s + (0.511 − 0.859i)21-s + 0.794i·23-s − 0.625·25-s + (−0.991 − 0.130i)27-s + (−0.649 − 0.375i)29-s + (0.184 + 0.106i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.636 + 0.771i$
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: \((2,\ 504,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82072 - 0.858789i\)
\(L(\frac12)\) \(\approx\) \(1.82072 - 0.858789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.930 + 1.46i)T \)
7 \( 1 + (-2.64 + 0.0810i)T \)
good5 \( 1 - 1.36T + 5T^{2} \)
11 \( 1 - 1.20iT - 11T^{2} \)
13 \( 1 + (-0.639 + 0.369i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.693 + 1.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.81 - 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.81iT - 23T^{2} \)
29 \( 1 + (3.50 + 2.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.02 - 0.594i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.10 + 8.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.670 + 1.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.490 - 0.848i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.63 + 2.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.77 - 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.73 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.36 - 2.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.19 - 3.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.84iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.14 - 5.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.05 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 6.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.91691029099805003132865304275, −9.678389213994835297610288979922, −9.035879983315932807004391999127, −7.87003832202152417858644194788, −7.44042204022834396938848372586, −6.17136645496828338183496825585, −5.30724739494807464717574898782, −3.87918573112802128413467573410, −2.41380831694203354161354467181, −1.41133993255106644676072905741, 1.82982749506599279963717509189, 3.11608419032397303821916017255, 4.41344366601713667947764916299, 5.21564071750493894929961680283, 6.27732723046593952771214729639, 7.73830223339326057871443917185, 8.435464872777616963524841782615, 9.315617080371298703636650123900, 10.07314280824633458203059907192, 11.01370284431632641611693169719

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