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Properties

Label 2-513-171.106-c1-0-7
Degree $2$
Conductor $513$
Sign $0.408 - 0.912i$
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  + (0.888 + 1.53i)2-s + (−0.580 + 1.00i)4-s + 1.27·5-s + (0.657 − 1.13i)7-s + 1.49·8-s + (1.13 + 1.97i)10-s + (0.130 − 0.225i)11-s + (0.933 − 1.61i)13-s + 2.33·14-s + (2.48 + 4.30i)16-s + (0.0508 − 0.0880i)17-s + (3.11 + 3.05i)19-s + (−0.742 + 1.28i)20-s + 0.462·22-s + (−0.611 + 1.05i)23-s + ⋯
L(s)  = 1  + (0.628 + 1.08i)2-s + (−0.290 + 0.502i)4-s + 0.572·5-s + (0.248 − 0.430i)7-s + 0.527·8-s + (0.359 + 0.623i)10-s + (0.0392 − 0.0679i)11-s + (0.259 − 0.448i)13-s + 0.625·14-s + (0.621 + 1.07i)16-s + (0.0123 − 0.0213i)17-s + (0.713 + 0.700i)19-s + (−0.166 + 0.287i)20-s + 0.0986·22-s + (−0.127 + 0.221i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98797 + 1.28793i\)
\(L(\frac12)\) \(\approx\) \(1.98797 + 1.28793i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 - 3.05i)T \)
good2 \( 1 + (-0.888 - 1.53i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.27T + 5T^{2} \)
7 \( 1 + (-0.657 + 1.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.130 + 0.225i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.933 + 1.61i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0508 + 0.0880i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.611 - 1.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.52T + 29T^{2} \)
31 \( 1 + (0.617 + 1.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.59T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 + (1.53 + 2.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.58T + 47T^{2} \)
53 \( 1 + (-2.59 - 4.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.02T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-0.390 + 0.677i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.19 + 14.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.397 - 0.687i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.03 + 5.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.75 + 9.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.83 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96083399625455292290041982521, −10.17504355183641106684442046508, −9.229274762424882108317847493726, −7.896457521815840566832917785766, −7.45788564602679377373839541669, −6.19951828815546076405363275473, −5.68745090114921877484042922015, −4.65464619799357487497331425665, −3.52377446725838405640356400171, −1.64436049141184164141193567479, 1.59682531701772062748393917904, 2.60512718286710339521633223747, 3.77896368330567547705447132279, 4.86880302327711774249536402315, 5.77924503723137404525115762644, 7.02403380984589795464992573845, 8.118453790166650484317709132345, 9.314747375686805082906753956302, 9.951432701597852631897692604890, 11.05234694777101582905992519787

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