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Properties

Label 2-513-171.7-c1-0-17
Degree $2$
Conductor $513$
Sign $0.942 + 0.333i$
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  + 2.39·2-s + 3.75·4-s + (0.359 − 0.623i)5-s + (1.65 − 2.87i)7-s + 4.21·8-s + (0.863 − 1.49i)10-s + (−0.550 + 0.953i)11-s − 4.74·13-s + (3.97 − 6.89i)14-s + 2.60·16-s + (3.13 + 5.42i)17-s + (−4.19 − 1.19i)19-s + (1.35 − 2.34i)20-s + (−1.32 + 2.28i)22-s + 2.22·23-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.87·4-s + (0.160 − 0.278i)5-s + (0.626 − 1.08i)7-s + 1.49·8-s + (0.273 − 0.472i)10-s + (−0.165 + 0.287i)11-s − 1.31·13-s + (1.06 − 1.84i)14-s + 0.650·16-s + (0.760 + 1.31i)17-s + (−0.961 − 0.274i)19-s + (0.302 − 0.523i)20-s + (−0.281 + 0.487i)22-s + 0.464·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.66157 - 0.628720i\)
\(L(\frac12)\) \(\approx\) \(3.66157 - 0.628720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.19 + 1.19i)T \)
good2 \( 1 - 2.39T + 2T^{2} \)
5 \( 1 + (-0.359 + 0.623i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.65 + 2.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.550 - 0.953i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.74T + 13T^{2} \)
17 \( 1 + (-3.13 - 5.42i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 - 2.22T + 23T^{2} \)
29 \( 1 + (-2.97 - 5.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.763 + 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.69T + 37T^{2} \)
41 \( 1 + (-2.84 + 4.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 + (-0.141 - 0.245i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.90 - 3.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.68 - 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.94 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 7.44T + 67T^{2} \)
71 \( 1 + (5.51 + 9.55i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.22 + 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + (-5.05 + 8.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.23 - 7.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−10.79290665106259573552852994012, −10.56195758841361880468885355129, −9.086336431456911329794249377401, −7.69155123477777562981107540092, −7.06244631296571644374562375456, −5.93434062759321880902822493347, −4.89078186605361381484755551709, −4.36981000193436081959082338356, −3.20288849908302621960483028220, −1.77298533069256942090321005326, 2.32582407956775116144459237892, 2.93947184128104211019140208323, 4.50779688132651365470642573418, 5.16298519450111117813084508080, 5.98581635066362071860183922311, 6.96664557396904254527422794708, 8.003466921989142370873781739065, 9.212871619719001034652335676286, 10.34843031480802954390648635022, 11.37687284431638478795316648747

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