Properties

Label 2-507-39.20-c1-0-4
Degree $2$
Conductor $507$
Sign $0.770 - 0.637i$
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  + (−0.389 − 1.45i)2-s + (−1.60 + 0.650i)3-s + (−0.232 + 0.133i)4-s + (−1.06 + 1.06i)5-s + (1.57 + 2.08i)6-s + (1.36 + 0.366i)7-s + (−1.84 − 1.84i)8-s + (2.15 − 2.08i)9-s + (1.96 + 1.13i)10-s + (−3.97 + 1.06i)11-s + (0.285 − 0.366i)12-s − 2.12i·14-s + (1.01 − 2.40i)15-s + (−2.23 + 3.86i)16-s + (2.51 + 4.36i)17-s + (−3.87 − 2.31i)18-s + ⋯
L(s)  = 1  + (−0.275 − 1.02i)2-s + (−0.926 + 0.375i)3-s + (−0.116 + 0.0669i)4-s + (−0.476 + 0.476i)5-s + (0.641 + 0.849i)6-s + (0.516 + 0.138i)7-s + (−0.652 − 0.652i)8-s + (0.717 − 0.696i)9-s + (0.621 + 0.358i)10-s + (−1.19 + 0.321i)11-s + (0.0823 − 0.105i)12-s − 0.569i·14-s + (0.262 − 0.620i)15-s + (−0.558 + 0.966i)16-s + (0.611 + 1.05i)17-s + (−0.913 − 0.546i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583240 + 0.210218i\)
\(L(\frac12)\) \(\approx\) \(0.583240 + 0.210218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 0.650i)T \)
13 \( 1 \)
good2 \( 1 + (0.389 + 1.45i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.06 - 1.06i)T - 5iT^{2} \)
7 \( 1 + (-1.36 - 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.97 - 1.06i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.51 - 4.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 3.73i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.20 - 3.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.46 - 2.46i)T + 31iT^{2} \)
37 \( 1 + (1.40 + 5.23i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.45 - 5.42i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.90 - 1.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (0.779 - 2.90i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.73 - 1.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.90 + 0.779i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.901 + 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (2.90 - 2.90i)T - 83iT^{2} \)
89 \( 1 + (9.01 - 2.41i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.437 + 1.63i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.80668022796267787339729290524, −10.49249063349910972402124186439, −9.765035524084433110121957105508, −8.443725845929031336804424083850, −7.39218202767366377102452208837, −6.28765379002825410183063835434, −5.32216420738861386683916184158, −4.09782990756930129654610203537, −2.99148776621567670188964490390, −1.45933613688877365227724116931, 0.46761797611954147487836630463, 2.61369643401669024792439966885, 4.67282291465948754249829862707, 5.27697536381517706342535982889, 6.29907705224673839399771396696, 7.24825231943782379992376828192, 7.920791111327499803526562125973, 8.554261310357936386959850078510, 9.951352645016587988533307307607, 10.96405556038025678814657439805

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