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Properties

Label 2-513-171.106-c1-0-12
Degree $2$
Conductor $513$
Sign $-0.919 - 0.394i$
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  + (−1.19 − 2.07i)2-s + (−1.87 + 3.25i)4-s − 0.719·5-s + (1.65 − 2.87i)7-s + 4.21·8-s + (0.863 + 1.49i)10-s + (−0.550 + 0.953i)11-s + (2.37 − 4.11i)13-s − 7.95·14-s + (−1.30 − 2.25i)16-s + (3.13 − 5.42i)17-s + (−4.19 + 1.19i)19-s + (1.35 − 2.34i)20-s + 2.64·22-s + (−1.11 + 1.92i)23-s + ⋯
L(s)  = 1  + (−0.848 − 1.46i)2-s + (−0.939 + 1.62i)4-s − 0.321·5-s + (0.626 − 1.08i)7-s + 1.49·8-s + (0.273 + 0.472i)10-s + (−0.165 + 0.287i)11-s + (0.658 − 1.14i)13-s − 2.12·14-s + (−0.325 − 0.563i)16-s + (0.760 − 1.31i)17-s + (−0.961 + 0.274i)19-s + (0.302 − 0.523i)20-s + 0.563·22-s + (−0.232 + 0.402i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.919 - 0.394i$
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: \((2,\ 513,\ (\ :1/2),\ -0.919 - 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127395 + 0.620158i\)
\(L(\frac12)\) \(\approx\) \(0.127395 + 0.620158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.19 - 1.19i)T \)
good2 \( 1 + (1.19 + 2.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.719T + 5T^{2} \)
7 \( 1 + (-1.65 + 2.87i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.550 - 0.953i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.37 + 4.11i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.13 + 5.42i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.11 - 1.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.94T + 29T^{2} \)
31 \( 1 + (0.763 + 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.69T + 37T^{2} \)
41 \( 1 + 5.68T + 41T^{2} \)
43 \( 1 + (2.30 + 3.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.283T + 47T^{2} \)
53 \( 1 + (1.90 + 3.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + (-3.72 + 6.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.51 - 9.55i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.22 - 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.11 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.05 + 8.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.23 + 7.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.16 + 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24945050309424641577830858371, −10.00165068376213644983731022279, −8.706575694721453581831490023230, −7.946251527343222313633018045129, −7.25556055951134661601636109442, −5.46236354352341384359498514961, −4.08907647756601607580819471707, −3.32515862367269244231219492989, −1.84744990975417270751007174750, −0.50762312102280784906709052084, 1.82541710788215260832513484295, 3.95102012701205925042072827724, 5.30258412750392632139776099849, 6.03985853219065585597567298600, 6.86019648853972295057564276615, 8.087207265286063851197778233746, 8.451807083942088611396060689532, 9.172228468750885845961578173107, 10.22862459802922108932947369857, 11.27528156424404605070276834884

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