Properties

Label 2-507-169.127-c1-0-28
Degree $2$
Conductor $507$
Sign $0.579 + 0.814i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (1.92 − 0.0776i)2-s + (−0.278 − 0.960i)3-s + (1.71 − 0.138i)4-s + (3.62 − 1.37i)5-s + (−0.610 − 1.82i)6-s + (1.17 − 2.76i)7-s + (−0.536 + 0.0651i)8-s + (−0.845 + 0.534i)9-s + (6.88 − 2.93i)10-s + (−1.78 + 2.82i)11-s + (−0.609 − 1.60i)12-s + (−2.71 + 2.37i)13-s + (2.05 − 5.42i)14-s + (−2.33 − 3.10i)15-s + (−4.42 + 0.718i)16-s + (−2.15 − 0.918i)17-s + ⋯
L(s)  = 1  + (1.36 − 0.0549i)2-s + (−0.160 − 0.554i)3-s + (0.857 − 0.0691i)4-s + (1.62 − 0.615i)5-s + (−0.249 − 0.746i)6-s + (0.445 − 1.04i)7-s + (−0.189 + 0.0230i)8-s + (−0.281 + 0.178i)9-s + (2.17 − 0.927i)10-s + (−0.538 + 0.851i)11-s + (−0.176 − 0.464i)12-s + (−0.752 + 0.659i)13-s + (0.549 − 1.45i)14-s + (−0.601 − 0.800i)15-s + (−1.10 + 0.179i)16-s + (−0.522 − 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.579 + 0.814i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.82715 - 1.45879i\)
\(L(\frac12)\) \(\approx\) \(2.82715 - 1.45879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.278 + 0.960i)T \)
13 \( 1 + (2.71 - 2.37i)T \)
good2 \( 1 + (-1.92 + 0.0776i)T + (1.99 - 0.160i)T^{2} \)
5 \( 1 + (-3.62 + 1.37i)T + (3.74 - 3.31i)T^{2} \)
7 \( 1 + (-1.17 + 2.76i)T + (-4.84 - 5.04i)T^{2} \)
11 \( 1 + (1.78 - 2.82i)T + (-4.71 - 9.93i)T^{2} \)
17 \( 1 + (2.15 + 0.918i)T + (11.7 + 12.2i)T^{2} \)
19 \( 1 + (-1.03 - 0.596i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.67 - 8.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.242 + 6.02i)T + (-28.9 + 2.33i)T^{2} \)
31 \( 1 + (-0.596 + 0.673i)T + (-3.73 - 30.7i)T^{2} \)
37 \( 1 + (-9.40 - 1.91i)T + (34.0 + 14.5i)T^{2} \)
41 \( 1 + (3.14 - 0.911i)T + (34.6 - 21.9i)T^{2} \)
43 \( 1 + (-1.20 - 5.91i)T + (-39.5 + 16.8i)T^{2} \)
47 \( 1 + (-0.455 - 0.314i)T + (16.6 + 43.9i)T^{2} \)
53 \( 1 + (0.0637 + 0.524i)T + (-51.4 + 12.6i)T^{2} \)
59 \( 1 + (-0.493 + 3.03i)T + (-55.9 - 18.6i)T^{2} \)
61 \( 1 + (11.0 + 8.27i)T + (16.9 + 58.5i)T^{2} \)
67 \( 1 + (0.412 - 5.11i)T + (-66.1 - 10.7i)T^{2} \)
71 \( 1 + (-3.47 - 3.34i)T + (2.85 + 70.9i)T^{2} \)
73 \( 1 + (7.24 + 13.7i)T + (-41.4 + 60.0i)T^{2} \)
79 \( 1 + (6.74 - 9.77i)T + (-28.0 - 73.8i)T^{2} \)
83 \( 1 + (1.18 + 4.79i)T + (-73.4 + 38.5i)T^{2} \)
89 \( 1 + (2.83 - 1.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.01 + 2.46i)T + (19.4 - 95.0i)T^{2} \)
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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

−11.06551210096305654494358999949, −9.809988701281211103588573477161, −9.304383209206517567263996838797, −7.73101133773754826827712268080, −6.82438343670001139605101786255, −5.89222744352954441900318392441, −4.99590678419408460950341757249, −4.43375845089897513611517068412, −2.64601753095198935963297261587, −1.58626527200680112678482977206, 2.49487960223716355929726404357, 2.94485045227885427465677863873, 4.69545981511298652042585277078, 5.46240294391148702319555888250, 5.90063371614032550528141192849, 6.86294749896899544111427704440, 8.631105516964525529143235458221, 9.290102723559804868637651344958, 10.45015977172199239401428130344, 10.97660947018334360946695121663

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