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Properties

Label 2-513-171.106-c1-0-16
Degree $2$
Conductor $513$
Sign $0.352 + 0.935i$
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  + (−0.0973 − 0.168i)2-s + (0.981 − 1.69i)4-s + 1.90·5-s + (1.69 − 2.93i)7-s − 0.771·8-s + (−0.185 − 0.321i)10-s + (0.311 − 0.539i)11-s + (−1.84 + 3.19i)13-s − 0.659·14-s + (−1.88 − 3.26i)16-s + (3.04 − 5.27i)17-s + (−1.14 + 4.20i)19-s + (1.86 − 3.23i)20-s − 0.121·22-s + (−3.92 + 6.79i)23-s + ⋯
L(s)  = 1  + (−0.0688 − 0.119i)2-s + (0.490 − 0.849i)4-s + 0.852·5-s + (0.640 − 1.10i)7-s − 0.272·8-s + (−0.0586 − 0.101i)10-s + (0.0939 − 0.162i)11-s + (−0.511 + 0.886i)13-s − 0.176·14-s + (−0.471 − 0.817i)16-s + (0.739 − 1.28i)17-s + (−0.262 + 0.964i)19-s + (0.418 − 0.724i)20-s − 0.0258·22-s + (−0.818 + 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.352 + 0.935i$
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: \((2,\ 513,\ (\ :1/2),\ 0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48632 - 1.02794i\)
\(L(\frac12)\) \(\approx\) \(1.48632 - 1.02794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 4.20i)T \)
good2 \( 1 + (0.0973 + 0.168i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.90T + 5T^{2} \)
7 \( 1 + (-1.69 + 2.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.311 + 0.539i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 3.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.04 + 5.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.92 - 6.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 + (0.910 + 1.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.63T + 37T^{2} \)
41 \( 1 - 4.03T + 41T^{2} \)
43 \( 1 + (2.54 + 4.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + (1.93 + 3.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.50T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + (0.523 - 0.905i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.56 - 2.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.06 + 3.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.16 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.35 - 9.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.25 - 9.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.34 + 12.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64890577974441954932246499150, −9.775929132081360232886558691336, −9.442535003424506988415657462622, −7.80217260279603024283659560692, −7.10141166702603960823706835500, −6.00309396668030496434815489120, −5.22958479677847086057931657441, −3.99739220610751644387298450144, −2.27529621845277049941123367163, −1.21675389392027080916408693321, 2.03767969362108931940225035108, 2.83970094792336167587352267380, 4.41764588753719038393583377643, 5.69958295406701083566621339070, 6.31167990261563930393525773430, 7.64026343658063165617554543203, 8.332653538680078847760505321254, 9.124412685760254033171772842075, 10.24005019483927197559416723160, 11.03613103843676781622790762091

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