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Properties

Label 2-507-13.4-c1-0-5
Degree $2$
Conductor $507$
Sign $0.964 + 0.265i$
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  + (−1.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)6-s + (−3 + 1.73i)7-s + 1.73i·8-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s + 12-s + 6·14-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s + 1.73i·18-s + (3 − 1.73i)19-s + 3.46i·21-s + (−3 − 5.19i)22-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s + (−0.612 + 0.353i)6-s + (−1.13 + 0.654i)7-s + 0.612i·8-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s + 0.288·12-s + 1.60·14-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s + 0.408i·18-s + (0.688 − 0.397i)19-s + 0.755i·21-s + (−0.639 − 1.10i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757699 - 0.102225i\)
\(L(\frac12)\) \(\approx\) \(0.757699 - 0.102225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-9 + 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9 - 5.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12 - 6.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.70823101381231867160946177586, −9.745189651565705237300685453805, −9.222001013497993981012487905346, −8.521584955690817098320288158146, −7.43514340593631472008992573431, −6.45756177055693616703738269197, −5.42793168713795955413936074046, −3.62528122291624945926772322282, −2.50052401206649616738105829340, −1.21799310853268573511548543471, 0.75791641788707269968501172580, 3.19004994129091839680150281350, 3.95810825017970346184274956770, 5.56565213639817030126170087840, 6.77056571878159983454859394878, 7.29663959649544371009586103992, 8.424945101879720860170709452951, 9.238730487837506726398540269362, 9.784279867347200058000851551102, 10.45737014804363880044918247950

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