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Properties

Label 2-507-13.10-c1-0-16
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  + (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

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

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: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46408 - 0.660812i\)
\(L(\frac12)\) \(\approx\) \(2.46408 - 0.660812i\)
\(L(\frac{3}{2})\) 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

−11.24593982952162065672997807376, −10.53667445395985412069068352846, −8.766412767882081559770776957051, −8.343942288379949006269209766103, −6.81648200709019426104055601472, −6.09647407429817606477198314280, −5.00768381866624460444620512322, −4.08592491066324910727497412636, −2.79909527343514097926599732398, −1.75015005353537646477471910523, 1.47016503720369968930984849513, 3.85954778405809703046999705220, 4.41407284435719193800350509795, 5.08483444940501980741308513408, 6.21285030045529491550335893905, 7.03835126186289396860847883205, 8.176051779666973369414069804420, 9.188874319032676380189237386904, 10.17009448612299849183233119958, 11.08220662965632001401708800141

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