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Properties

Label 2-513-171.164-c1-0-13
Degree $2$
Conductor $513$
Sign $0.996 + 0.0889i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + 2.03·2-s + 2.14·4-s + (0.682 + 0.394i)5-s + (1.40 − 2.44i)7-s + 0.290·8-s + (1.38 + 0.801i)10-s + (3.83 + 2.21i)11-s + 2.47i·13-s + (2.86 − 4.96i)14-s − 3.69·16-s + (3.48 − 2.00i)17-s + (−3.58 − 2.48i)19-s + (1.46 + 0.844i)20-s + (7.79 + 4.50i)22-s + 1.48i·23-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.07·4-s + (0.305 + 0.176i)5-s + (0.532 − 0.922i)7-s + 0.102·8-s + (0.439 + 0.253i)10-s + (1.15 + 0.666i)11-s + 0.687i·13-s + (0.766 − 1.32i)14-s − 0.923·16-s + (0.844 − 0.487i)17-s + (−0.822 − 0.569i)19-s + (0.326 + 0.188i)20-s + (1.66 + 0.959i)22-s + 0.309i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.996 + 0.0889i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.996 + 0.0889i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.22342 - 0.143704i\)
\(L(\frac12)\) \(\approx\) \(3.22342 - 0.143704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (3.58 + 2.48i)T \)
good2 \( 1 - 2.03T + 2T^{2} \)
5 \( 1 + (-0.682 - 0.394i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.40 + 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.83 - 2.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.47iT - 13T^{2} \)
17 \( 1 + (-3.48 + 2.00i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 - 1.48iT - 23T^{2} \)
29 \( 1 + (0.371 + 0.643i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.821 + 0.474i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.15iT - 37T^{2} \)
41 \( 1 + (5.31 - 9.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + (8.85 - 5.11i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.994 - 1.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.57 + 6.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.69 + 8.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + (-4.07 - 7.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.16 + 8.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.608iT - 79T^{2} \)
83 \( 1 + (10.2 + 5.91i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.203 + 0.352i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−11.32664756540699439960251250067, −10.08339490560761329131897055247, −9.284348644843058193821947174198, −7.947429920120544924262879440668, −6.76877819481913122647594530636, −6.31720286814482629801368994257, −4.84324804313850784611299063019, −4.36153919606001334568773967083, −3.27055355343701463894917475363, −1.74580094223284872178572460105, 1.86002817949535326505877078839, 3.26614993601204285313578457022, 4.15246932169804791514794602235, 5.54602570441952333095994988328, 5.70632304191531200815541145621, 6.87111824711250842602897167271, 8.344644397705813508803969228954, 8.957261296451522923589613011879, 10.19164115092501056947909047508, 11.32583357168004112940889420549

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