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Properties

Label 2-504-168.83-c2-0-5
Degree $2$
Conductor $504$
Sign $-0.974 + 0.225i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.852 + 1.80i)2-s + (−2.54 + 3.08i)4-s − 9.10i·5-s + (−0.958 + 6.93i)7-s + (−7.75 − 1.98i)8-s + (16.4 − 7.76i)10-s + 9.66i·11-s − 2.31·13-s + (−13.3 + 4.17i)14-s + (−3.02 − 15.7i)16-s − 21.8·17-s + 26.9i·19-s + (28.0 + 23.2i)20-s + (−17.4 + 8.23i)22-s + 15.4·23-s + ⋯
L(s)  = 1  + (0.426 + 0.904i)2-s + (−0.636 + 0.770i)4-s − 1.82i·5-s + (−0.136 + 0.990i)7-s + (−0.968 − 0.247i)8-s + (1.64 − 0.776i)10-s + 0.878i·11-s − 0.178·13-s + (−0.954 + 0.298i)14-s + (−0.188 − 0.982i)16-s − 1.28·17-s + 1.41i·19-s + (1.40 + 1.16i)20-s + (−0.794 + 0.374i)22-s + 0.670·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ -0.974 + 0.225i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6479609603\)
\(L(\frac12)\) \(\approx\) \(0.6479609603\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.852 - 1.80i)T \)
3 \( 1 \)
7 \( 1 + (0.958 - 6.93i)T \)
good5 \( 1 + 9.10iT - 25T^{2} \)
11 \( 1 - 9.66iT - 121T^{2} \)
13 \( 1 + 2.31T + 169T^{2} \)
17 \( 1 + 21.8T + 289T^{2} \)
19 \( 1 - 26.9iT - 361T^{2} \)
23 \( 1 - 15.4T + 529T^{2} \)
29 \( 1 + 18.1T + 841T^{2} \)
31 \( 1 + 45.9T + 961T^{2} \)
37 \( 1 - 32.3iT - 1.36e3T^{2} \)
41 \( 1 + 33.0T + 1.68e3T^{2} \)
43 \( 1 + 60.5T + 1.84e3T^{2} \)
47 \( 1 - 29.4iT - 2.20e3T^{2} \)
53 \( 1 - 65.7T + 2.80e3T^{2} \)
59 \( 1 - 8.47T + 3.48e3T^{2} \)
61 \( 1 - 74.9T + 3.72e3T^{2} \)
67 \( 1 - 20.2T + 4.48e3T^{2} \)
71 \( 1 - 21.7T + 5.04e3T^{2} \)
73 \( 1 + 100. iT - 5.32e3T^{2} \)
79 \( 1 + 78.9iT - 6.24e3T^{2} \)
83 \( 1 + 67.6T + 6.88e3T^{2} \)
89 \( 1 - 130.T + 7.92e3T^{2} \)
97 \( 1 - 175. iT - 9.40e3T^{2} \)
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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

−11.67709214957590188880495547955, −9.850427428919510525277797539868, −9.050276791500745122108766332800, −8.581766901953905017934505425855, −7.66401312605924276450375781320, −6.44568700215539578403322582584, −5.36564682224110188302921883840, −4.88729617972409543022443090004, −3.80760912270126650434965306146, −1.90315929452287505580657978013, 0.21145126605239024723015253536, 2.23397369699630200340349267653, 3.24005879679490419313479969046, 4.00197821689066731938660523003, 5.41180768668185742587726660277, 6.71378354658168804689003121642, 7.08633433259458998666346561333, 8.667823036488066023268666548185, 9.715498978242327949011060491726, 10.58507312516574807822072347539

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