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Properties

Label 2-504-56.53-c1-0-18
Degree $2$
Conductor $504$
Sign $0.316 + 0.948i$
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  + (−0.867 + 1.11i)2-s + (−0.496 − 1.93i)4-s + (−2.93 + 1.69i)5-s + (−1.85 + 1.88i)7-s + (2.59 + 1.12i)8-s + (0.652 − 4.74i)10-s + (0.0932 + 0.0538i)11-s − 1.50i·13-s + (−0.504 − 3.70i)14-s + (−3.50 + 1.92i)16-s + (0.214 − 0.372i)17-s + (4.32 − 2.49i)19-s + (4.73 + 4.84i)20-s + (−0.141 + 0.0575i)22-s + (−4.56 − 7.90i)23-s + ⋯
L(s)  = 1  + (−0.613 + 0.789i)2-s + (−0.248 − 0.968i)4-s + (−1.31 + 0.757i)5-s + (−0.700 + 0.713i)7-s + (0.917 + 0.398i)8-s + (0.206 − 1.50i)10-s + (0.0281 + 0.0162i)11-s − 0.416i·13-s + (−0.134 − 0.990i)14-s + (−0.876 + 0.480i)16-s + (0.0520 − 0.0902i)17-s + (0.992 − 0.573i)19-s + (1.05 + 1.08i)20-s + (−0.0300 + 0.0122i)22-s + (−0.951 − 1.64i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177759 - 0.128029i\)
\(L(\frac12)\) \(\approx\) \(0.177759 - 0.128029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.867 - 1.11i)T \)
3 \( 1 \)
7 \( 1 + (1.85 - 1.88i)T \)
good5 \( 1 + (2.93 - 1.69i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0932 - 0.0538i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.50iT - 13T^{2} \)
17 \( 1 + (-0.214 + 0.372i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.32 + 2.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.56 + 7.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.95iT - 29T^{2} \)
31 \( 1 + (0.393 - 0.680i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.68 - 4.43i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 + 6.65iT - 43T^{2} \)
47 \( 1 + (-2.87 - 4.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.286 - 0.165i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.63 - 4.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.76 + 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.79 + 1.61i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.72T + 71T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.785 + 1.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.33iT - 83T^{2} \)
89 \( 1 + (3.62 + 6.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−10.51566324492680142816034200784, −9.857918945755887084477007007532, −8.699818320080233207698942588278, −8.054305381440386660055374509985, −7.10174688672051445233686681092, −6.43063398577637997718309770159, −5.29550784973805049551844115047, −3.97476456861652622971427677980, −2.67208201001362048812581357466, −0.17214333770015165648542447230, 1.37339033117193819851098318986, 3.48084301078545562633091942831, 3.85604161523892774055619653539, 5.16963029138157883808859068845, 6.99632549850124128111491255106, 7.65030808024374377765706748716, 8.488465878876412805240505087492, 9.401050043079030261361676173480, 10.16175086900220401078079596281, 11.18716203618320213533854514602

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