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Properties

Label 2-513-171.7-c1-0-10
Degree $2$
Conductor $513$
Sign $-0.505 + 0.862i$
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  − 0.370·2-s − 1.86·4-s + (−1.77 + 3.07i)5-s + (−0.124 + 0.216i)7-s + 1.43·8-s + (0.659 − 1.14i)10-s + (0.815 − 1.41i)11-s − 1.32·13-s + (0.0463 − 0.0802i)14-s + 3.19·16-s + (−3.72 − 6.46i)17-s + (−4.07 + 1.54i)19-s + (3.31 − 5.73i)20-s + (−0.302 + 0.524i)22-s + 4.49·23-s + ⋯
L(s)  = 1  − 0.262·2-s − 0.931·4-s + (−0.794 + 1.37i)5-s + (−0.0471 + 0.0817i)7-s + 0.506·8-s + (0.208 − 0.361i)10-s + (0.245 − 0.426i)11-s − 0.367·13-s + (0.0123 − 0.0214i)14-s + 0.798·16-s + (−0.904 − 1.56i)17-s + (−0.935 + 0.354i)19-s + (0.740 − 1.28i)20-s + (−0.0645 + 0.111i)22-s + 0.936·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.505 + 0.862i$
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: \((2,\ 513,\ (\ :1/2),\ -0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0918244 - 0.160184i\)
\(L(\frac12)\) \(\approx\) \(0.0918244 - 0.160184i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.370T + 2T^{2} \)
5 \( 1 + (1.77 - 3.07i)T + (-2.5 - 4.33i)T^{2} \)
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 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 + (3.72 + 6.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 - 4.49T + 23T^{2} \)
29 \( 1 + (2.06 + 3.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.32 + 7.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.10T + 37T^{2} \)
41 \( 1 + (-2.77 + 4.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + (-1.68 - 2.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.254 - 0.440i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.23 - 9.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.07 + 3.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 0.799T + 67T^{2} \)
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 \( 1 - 9.85T + 79T^{2} \)
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 \( 1 - 6.43T + 97T^{2} \)
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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

−10.73000272710330492091016935513, −9.616113878388108107920340940448, −8.925010793595677064274110344504, −7.79280783440951606643429554782, −7.16667920540175555684009043946, −6.07793257912669038079156625677, −4.68826622572894053183648017168, −3.76278538982616355322403311243, −2.61656336010643432902476805826, −0.12795651883310362648492955313, 1.48600111759992999585205509991, 3.75941318327982882416136146036, 4.53688700272673774592163406564, 5.22612904667818002967915435803, 6.75552377600353047118992780151, 7.926373085686126180327690901143, 8.733211665356687714925471481532, 9.023201324001703561803995973524, 10.20931184278110990783945658044, 11.12061918968951749735878076863

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