Properties

Label 2-507-13.4-c1-0-9
Degree $2$
Conductor $507$
Sign $-0.207 + 0.978i$
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  + (−1.56 − 0.900i)2-s + (−0.5 + 0.866i)3-s + (0.623 + 1.07i)4-s + 1.44i·5-s + (1.56 − 0.900i)6-s + (−2.98 + 1.72i)7-s + 1.35i·8-s + (−0.499 − 0.866i)9-s + (1.30 − 2.25i)10-s + (−4.49 − 2.59i)11-s − 1.24·12-s + 6.20·14-s + (−1.25 − 0.722i)15-s + (2.46 − 4.27i)16-s + (−0.376 − 0.652i)17-s + 1.80i·18-s + ⋯
L(s)  = 1  + (−1.10 − 0.637i)2-s + (−0.288 + 0.499i)3-s + (0.311 + 0.539i)4-s + 0.646i·5-s + (0.637 − 0.367i)6-s + (−1.12 + 0.651i)7-s + 0.479i·8-s + (−0.166 − 0.288i)9-s + (0.411 − 0.713i)10-s + (−1.35 − 0.781i)11-s − 0.359·12-s + 1.65·14-s + (−0.323 − 0.186i)15-s + (0.617 − 1.06i)16-s + (−0.0913 − 0.158i)17-s + 0.424i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.207 + 0.978i$
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: \((2,\ 507,\ (\ :1/2),\ -0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198000 - 0.244481i\)
\(L(\frac12)\) \(\approx\) \(0.198000 - 0.244481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 + 0.900i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.44iT - 5T^{2} \)
7 \( 1 + (2.98 - 1.72i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.49 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.376 + 0.652i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.89 + 3.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.41 - 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.95 + 3.38i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + (-5.41 - 3.12i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.56 + 0.900i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.54 + 6.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 + (1.62 - 0.939i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.67 + 2.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 - 2.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.89 - 4.55i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.95iT - 73T^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 + 6.46iT - 83T^{2} \)
89 \( 1 + (-1.00 - 0.579i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.49 + 4.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.43979743715989082986777808361, −9.869340557792317176827965824936, −9.203120438948985050416373181914, −8.262892310366920756824991729634, −7.20365600768065254055096817972, −5.93728840872067988295654474471, −5.15950468799325675328820556119, −3.19717104586340876490429707256, −2.63226929964739974845627542132, −0.32504629097272838986612676365, 1.13256160462137661469720063768, 3.14992485777983304672731293756, 4.70980011831986984914031853630, 5.92356702650905268199807043643, 6.91382008058073015371233685020, 7.59196657999047100187956776272, 8.287232703229240312781056406990, 9.429693895005883574495883270364, 10.01442986304066325518476490204, 10.75098135827757529063740367458

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