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Properties

Label 2-513-171.106-c1-0-14
Degree $2$
Conductor $513$
Sign $0.954 + 0.299i$
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  + (0.185 + 0.321i)2-s + (0.931 − 1.61i)4-s + 3.55·5-s + (−0.124 + 0.216i)7-s + 1.43·8-s + (0.659 + 1.14i)10-s + (0.815 − 1.41i)11-s + (0.662 − 1.14i)13-s − 0.0926·14-s + (−1.59 − 2.76i)16-s + (−3.72 + 6.46i)17-s + (−4.07 − 1.54i)19-s + (3.31 − 5.73i)20-s + 0.605·22-s + (−2.24 + 3.88i)23-s + ⋯
L(s)  = 1  + (0.131 + 0.227i)2-s + (0.465 − 0.806i)4-s + 1.58·5-s + (−0.0471 + 0.0817i)7-s + 0.506·8-s + (0.208 + 0.361i)10-s + (0.245 − 0.426i)11-s + (0.183 − 0.318i)13-s − 0.0247·14-s + (−0.399 − 0.691i)16-s + (−0.904 + 1.56i)17-s + (−0.935 − 0.354i)19-s + (0.740 − 1.28i)20-s + 0.129·22-s + (−0.468 + 0.811i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.954 + 0.299i$
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: \((2,\ 513,\ (\ :1/2),\ 0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13684 - 0.327550i\)
\(L(\frac12)\) \(\approx\) \(2.13684 - 0.327550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.07 + 1.54i)T \)
good2 \( 1 + (-0.185 - 0.321i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.55T + 5T^{2} \)
7 \( 1 + (0.124 - 0.216i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.815 + 1.41i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.662 + 1.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.72 - 6.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.24 - 3.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + (4.32 + 7.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 + 5.54T + 41T^{2} \)
43 \( 1 + (-5.02 - 8.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.36T + 47T^{2} \)
53 \( 1 + (0.254 + 0.440i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 4.14T + 61T^{2} \)
67 \( 1 + (-0.399 + 0.692i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.60 - 9.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.84 - 3.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.92 + 8.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.185 - 0.320i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.01 + 6.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.21 + 5.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71050393139690323062603583206, −10.02725188189955783763040076272, −9.256995979067970769534402907956, −8.260844731273364766458275202802, −6.80153441442984995098614083048, −6.03916938519857439918135063537, −5.67929633836951825552183322502, −4.29793805114176773442183396752, −2.46428386264290316985153290573, −1.52904410817565480311669192779, 1.89681897150239608783829635075, 2.67035455269633195278729527137, 4.17109672517452225196943178380, 5.27610723850618772609322729122, 6.61258804715472538476938055833, 6.93486019406942855182648680249, 8.447278992968143442923059172810, 9.166968473856491700912488932807, 10.15993226088771360471916681490, 10.84501528845300853488653793865

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