↑

Properties

Label 2-513-171.164-c1-0-3
Degree $2$
Conductor $513$
Sign $-0.656 - 0.754i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.29·2-s + 3.25·4-s + (−0.0873 − 0.0504i)5-s + (−0.977 + 1.69i)7-s − 2.88·8-s + (0.200 + 0.115i)10-s + (0.427 + 0.246i)11-s − 0.0391i·13-s + (2.24 − 3.88i)14-s + 0.102·16-s + (−3.33 + 1.92i)17-s + (0.824 + 4.28i)19-s + (−0.284 − 0.164i)20-s + (−0.980 − 0.566i)22-s − 5.30i·23-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.62·4-s + (−0.0390 − 0.0225i)5-s + (−0.369 + 0.640i)7-s − 1.02·8-s + (0.0633 + 0.0365i)10-s + (0.128 + 0.0744i)11-s − 0.0108i·13-s + (0.599 − 1.03i)14-s + 0.0256·16-s + (−0.808 + 0.467i)17-s + (0.189 + 0.981i)19-s + (−0.0636 − 0.0367i)20-s + (−0.209 − 0.120i)22-s − 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.656 - 0.754i$
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: \((2,\ 513,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128812 + 0.282656i\)
\(L(\frac12)\) \(\approx\) \(0.128812 + 0.282656i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.824 - 4.28i)T \)
good2 \( 1 + 2.29T + 2T^{2} \)
5 \( 1 + (0.0873 + 0.0504i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.977 - 1.69i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.427 - 0.246i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.0391iT - 13T^{2} \)
17 \( 1 + (3.33 - 1.92i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + 5.30iT - 23T^{2} \)
29 \( 1 + (-3.27 - 5.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.634 + 0.366i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.10iT - 37T^{2} \)
41 \( 1 + (2.92 - 5.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + (8.31 - 4.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.01 - 6.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.33 - 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.11 + 1.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 0.750iT - 67T^{2} \)
71 \( 1 + (4.08 + 7.07i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.16 - 2.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 16.3iT - 79T^{2} \)
83 \( 1 + (-6.89 - 3.98i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.30 + 3.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.44iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.85306389400615890991010555190, −10.16538449168248114161567514190, −9.411776332074460200586805050831, −8.520842377870928942469496389382, −8.056547583303870998632380526456, −6.77685756378130302119454977411, −6.15078711893495527470260233095, −4.56626707050066047381738613873, −2.88378444774767700888725711371, −1.60082177799139441748658769204, 0.30906151910721334036085849810, 1.88904592956114986326301094055, 3.41046004159702356923390988792, 4.91444813266226114401686483702, 6.49324412241245843777843511059, 7.13680614397868097990203814072, 7.976661032878942667786741534851, 8.933121452122159656921649041704, 9.637212005898875749249286677457, 10.28473959961813557208040809074

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