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Properties

Label 2-507-13.4-c1-0-14
Degree $2$
Conductor $507$
Sign $-0.865 + 0.500i$
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  + (−2.33 − 1.34i)2-s + (0.5 − 0.866i)3-s + (2.62 + 4.54i)4-s − 1.04i·5-s + (−2.33 + 1.34i)6-s + (−0.480 + 0.277i)7-s − 8.74i·8-s + (−0.499 − 0.866i)9-s + (−1.41 + 2.44i)10-s + (2.52 + 1.45i)11-s + 5.24·12-s + 1.49·14-s + (−0.908 − 0.524i)15-s + (−6.51 + 11.2i)16-s + (−2.42 − 4.20i)17-s + 2.69i·18-s + ⋯
L(s)  = 1  + (−1.64 − 0.951i)2-s + (0.288 − 0.499i)3-s + (1.31 + 2.27i)4-s − 0.469i·5-s + (−0.951 + 0.549i)6-s + (−0.181 + 0.104i)7-s − 3.09i·8-s + (−0.166 − 0.288i)9-s + (−0.446 + 0.773i)10-s + (0.760 + 0.438i)11-s + 1.51·12-s + 0.399·14-s + (−0.234 − 0.135i)15-s + (−1.62 + 2.82i)16-s + (−0.588 − 1.01i)17-s + 0.634i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.865 + 0.500i$
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: \((2,\ 507,\ (\ :1/2),\ -0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161394 - 0.601817i\)
\(L(\frac12)\) \(\approx\) \(0.161394 - 0.601817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 + 1.34i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 1.04iT - 5T^{2} \)
7 \( 1 + (0.480 - 0.277i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.52 - 1.45i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.42 + 4.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.652 - 0.376i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.88 + 4.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.955 + 1.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.51iT - 31T^{2} \)
37 \( 1 + (-4.98 - 2.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.25 + 2.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.54 + 9.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.753iT - 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (3.54 - 2.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 - 2.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 0.936i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.09 - 5.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 2.64iT - 83T^{2} \)
89 \( 1 + (-8.59 - 4.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.7 + 8.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39474528596142390110300358028, −9.434211618798764597185200757695, −8.979199660628632044206288991036, −8.175291598060569646549683850999, −7.22184460292621091697304420238, −6.47277089795986290088001157218, −4.43099355285883108709303162742, −3.01051209939303876247034328200, −1.98622969129442623740589535926, −0.64492312885108499945280536978, 1.51025177081313103338233497784, 3.23375014122491870645571997169, 4.97552456095678876855048488895, 6.28042500133211928780701606466, 6.78150583435903008830309337169, 7.87430452050113637516013673426, 8.709691072770402764267228584486, 9.274240850936022821180951505849, 10.18431178632801406804241177938, 10.84796608143393060307737152856

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