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Properties

Label 2-504-56.37-c1-0-2
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: $\chi_{504} (37, \cdot )$
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

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

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