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Properties

Label 2-513-171.106-c1-0-1
Degree $2$
Conductor $513$
Sign $-0.406 - 0.913i$
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  + (0.0732 + 0.126i)2-s + (0.989 − 1.71i)4-s − 2.57·5-s + (−1.73 + 3.01i)7-s + 0.582·8-s + (−0.188 − 0.326i)10-s + (−2.07 + 3.60i)11-s + (−2.29 + 3.98i)13-s − 0.509·14-s + (−1.93 − 3.35i)16-s + (1.50 − 2.61i)17-s + (3.65 + 2.37i)19-s + (−2.54 + 4.40i)20-s − 0.609·22-s + (−2.45 + 4.25i)23-s + ⋯
L(s)  = 1  + (0.0518 + 0.0897i)2-s + (0.494 − 0.856i)4-s − 1.14·5-s + (−0.657 + 1.13i)7-s + 0.206·8-s + (−0.0595 − 0.103i)10-s + (−0.626 + 1.08i)11-s + (−0.637 + 1.10i)13-s − 0.136·14-s + (−0.483 − 0.838i)16-s + (0.365 − 0.633i)17-s + (0.838 + 0.544i)19-s + (−0.568 + 0.984i)20-s − 0.129·22-s + (−0.512 + 0.888i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.406 - 0.913i$
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: \((2,\ 513,\ (\ :1/2),\ -0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387010 + 0.596056i\)
\(L(\frac12)\) \(\approx\) \(0.387010 + 0.596056i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.65 - 2.37i)T \)
good2 \( 1 + (-0.0732 - 0.126i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.57T + 5T^{2} \)
7 \( 1 + (1.73 - 3.01i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.07 - 3.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.29 - 3.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.50 + 2.61i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.45 - 4.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.71T + 29T^{2} \)
31 \( 1 + (-3.31 - 5.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 + (2.39 + 4.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.71T + 47T^{2} \)
53 \( 1 + (5.35 + 9.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.96T + 59T^{2} \)
61 \( 1 - 0.944T + 61T^{2} \)
67 \( 1 + (0.688 - 1.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.45 - 7.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.14 - 3.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.69 + 2.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.52 - 7.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.96 - 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.930 + 1.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.49825897018477953750325873523, −9.951062010889983798826200777014, −9.760974808928854724998080679187, −8.468159137518396929778904990423, −7.34104769459524212214254240939, −6.76974405707143230471414703478, −5.48801007405975382508481283802, −4.73774242476177270308422077556, −3.21485502434086615358556793596, −1.96073663121268421932193980409, 0.39181759952514337580384015666, 2.97902671435115695637783606498, 3.51476390346158394962534305265, 4.61365338458788735447845468130, 6.16205072284301815522148217448, 7.23349637788111752448569772951, 7.888942057256975272664753689198, 8.394340760952391803208628715886, 10.03129004797098762494797095966, 10.67943713152476113940607272722

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