Properties

Label 2-507-39.11-c1-0-10
Degree $2$
Conductor $507$
Sign $0.878 - 0.477i$
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  + (−0.866 − 1.5i)3-s + (−1.73 − i)4-s + (1.13 + 4.23i)7-s + (−1.5 + 2.59i)9-s + 3.46i·12-s + (1.99 + 3.46i)16-s + (3.09 − 0.830i)19-s + (5.36 − 5.36i)21-s + 5i·25-s + 5.19·27-s + (2.26 − 8.46i)28-s + (0.830 + 0.830i)31-s + (5.19 − 3i)36-s + (11.5 + 3.09i)37-s + (1.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.866 − 0.5i)4-s + (0.428 + 1.59i)7-s + (−0.5 + 0.866i)9-s + 0.999i·12-s + (0.499 + 0.866i)16-s + (0.710 − 0.190i)19-s + (1.17 − 1.17i)21-s + i·25-s + 1.00·27-s + (0.428 − 1.59i)28-s + (0.149 + 0.149i)31-s + (0.866 − 0.5i)36-s + (1.90 + 0.509i)37-s + (0.228 + 0.132i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.858874 + 0.218304i\)
\(L(\frac12)\) \(\approx\) \(0.858874 + 0.218304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (1.73 + i)T^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (-1.13 - 4.23i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.09 + 0.830i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 0.830i)T + 31iT^{2} \)
37 \( 1 + (-11.5 - 3.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.33 - 7.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.23 - 15.7i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.63 - 7.63i)T - 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.59 + 2.57i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.28221717820999632676928431538, −10.01753666914375782051150441697, −9.056255529254453960244709871150, −8.411975288773671278822807134213, −7.40513329792198021013826116517, −6.02411790243277064400266106181, −5.54991058153320351694422923174, −4.63997093429983020358734671442, −2.73470212527227950490945767540, −1.34884139883707453633209073192, 0.66203109531591936146532843532, 3.31909925986153169362519755837, 4.26394308657462832395167516331, 4.77091360016518354446522879715, 6.06601186471511997225804557721, 7.37976779402269249295191079346, 8.133378161777257086445646289290, 9.273199851842492720526283350444, 9.985844880291754014238837379987, 10.72426839992464117607540838651

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