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Properties

Label 2-513-171.106-c1-0-17
Degree $2$
Conductor $513$
Sign $-0.432 - 0.901i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s − 3·5-s + (−0.5 + 0.866i)7-s − 3·8-s + (1.5 + 2.59i)10-s + (2.5 − 4.33i)11-s + (−1 + 1.73i)13-s + 0.999·14-s + (0.500 + 0.866i)16-s + (−2.5 + 4.33i)17-s + (−4 + 1.73i)19-s + (−1.50 + 2.59i)20-s − 5·22-s + (−4 + 6.92i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s − 1.34·5-s + (−0.188 + 0.327i)7-s − 1.06·8-s + (0.474 + 0.821i)10-s + (0.753 − 1.30i)11-s + (−0.277 + 0.480i)13-s + 0.267·14-s + (0.125 + 0.216i)16-s + (−0.606 + 1.05i)17-s + (−0.917 + 0.397i)19-s + (−0.335 + 0.580i)20-s − 1.06·22-s + (−0.834 + 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.432 - 0.901i$
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: \(1\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.432 - 0.901i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58638783892411397153710344868, −9.331299090685136766990487254915, −8.689371396148336096517982481822, −7.77474724790656901848344760231, −6.49386119934432030736135466133, −5.80067242912676175172683977763, −4.12661715272118387976707637731, −3.34769565490386540788898149734, −1.77239612631088751425230019038, 0, 2.59067895697873275870111657084, 3.92082924039623717439462001031, 4.68923183829126563639817132496, 6.50485847286019040085541031313, 7.04803389212843103254945063043, 7.79775419309360007046617500826, 8.596419359294293468490418897996, 9.500109733327351599416080101829, 10.66476065981384377796485474259

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