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Properties

Label 2-507-13.12-c3-0-15
Degree $2$
Conductor $507$
Sign $0.277 + 0.960i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.04i·2-s + 3·3-s − 17.4·4-s + 20.1i·5-s + 15.1i·6-s + 15.4i·7-s − 47.7i·8-s + 9·9-s − 101.·10-s + 26.9i·11-s − 52.3·12-s − 77.8·14-s + 60.3i·15-s + 101.·16-s − 23.2·17-s + 45.4i·18-s + ⋯
L(s)  = 1  + 1.78i·2-s + 0.577·3-s − 2.18·4-s + 1.79i·5-s + 1.02i·6-s + 0.833i·7-s − 2.10i·8-s + 0.333·9-s − 3.20·10-s + 0.738i·11-s − 1.25·12-s − 1.48·14-s + 1.03i·15-s + 1.57·16-s − 0.331·17-s + 0.594i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.778779226\)
\(L(\frac12)\) \(\approx\) \(1.778779226\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 - 5.04iT - 8T^{2} \)
5 \( 1 - 20.1iT - 125T^{2} \)
7 \( 1 - 15.4iT - 343T^{2} \)
11 \( 1 - 26.9iT - 1.33e3T^{2} \)
17 \( 1 + 23.2T + 4.91e3T^{2} \)
19 \( 1 - 45.0iT - 6.85e3T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 - 2.29T + 2.43e4T^{2} \)
31 \( 1 - 37.7iT - 2.97e4T^{2} \)
37 \( 1 - 313. iT - 5.06e4T^{2} \)
41 \( 1 - 5.86iT - 6.89e4T^{2} \)
43 \( 1 - 360.T + 7.95e4T^{2} \)
47 \( 1 + 209. iT - 1.03e5T^{2} \)
53 \( 1 - 276.T + 1.48e5T^{2} \)
59 \( 1 + 543. iT - 2.05e5T^{2} \)
61 \( 1 - 205.T + 2.26e5T^{2} \)
67 \( 1 + 492. iT - 3.00e5T^{2} \)
71 \( 1 + 826. iT - 3.57e5T^{2} \)
73 \( 1 - 66.1iT - 3.89e5T^{2} \)
79 \( 1 - 317.T + 4.93e5T^{2} \)
83 \( 1 + 141. iT - 5.71e5T^{2} \)
89 \( 1 - 641. iT - 7.04e5T^{2} \)
97 \( 1 - 1.11e3iT - 9.12e5T^{2} \)
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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

−10.98721639827192056240419208337, −9.962458357635676365702489825620, −9.182433981711993330149107620739, −8.232103936682628922357630412443, −7.36530194169441714296074720799, −6.78726443307698489591514510169, −6.03626102668053428397169054015, −4.89077082101052721493153328947, −3.56493090612872768780951254049, −2.38752970530093312358099007578, 0.58775642677147334099330409856, 1.19768029427825299343967438789, 2.55647787616211816084999485104, 3.87169238217435153900617374955, 4.45583606469637489463359484700, 5.47157255189814138622266079170, 7.44109405739139011769846925553, 8.725108501356273107639849011138, 8.907000863523226801501257261765, 9.797864905826911584959760249603

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