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Properties

Label 2-513-171.106-c1-0-6
Degree $2$
Conductor $513$
Sign $0.0330 + 0.999i$
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  + (−1.01 − 1.74i)2-s + (−1.04 + 1.80i)4-s + 4.18·5-s + (−0.976 + 1.69i)7-s + 0.164·8-s + (−4.22 − 7.32i)10-s + (0.669 − 1.15i)11-s + (−0.975 + 1.68i)13-s + 3.94·14-s + (1.91 + 3.31i)16-s + (3.34 − 5.78i)17-s + (4.11 − 1.44i)19-s + (−4.35 + 7.54i)20-s − 2.70·22-s + (−0.986 + 1.70i)23-s + ⋯
L(s)  = 1  + (−0.714 − 1.23i)2-s + (−0.520 + 0.901i)4-s + 1.87·5-s + (−0.368 + 0.639i)7-s + 0.0583·8-s + (−1.33 − 2.31i)10-s + (0.201 − 0.349i)11-s + (−0.270 + 0.468i)13-s + 1.05·14-s + (0.478 + 0.829i)16-s + (0.810 − 1.40i)17-s + (0.943 − 0.331i)19-s + (−0.974 + 1.68i)20-s − 0.576·22-s + (−0.205 + 0.356i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0330 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0330 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.0330 + 0.999i$
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: \((2,\ 513,\ (\ :1/2),\ 0.0330 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941276 - 0.910680i\)
\(L(\frac12)\) \(\approx\) \(0.941276 - 0.910680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 + 1.44i)T \)
good2 \( 1 + (1.01 + 1.74i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 4.18T + 5T^{2} \)
7 \( 1 + (0.976 - 1.69i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.669 + 1.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.975 - 1.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.34 + 5.78i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.986 - 1.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.90T + 29T^{2} \)
31 \( 1 + (0.385 + 0.668i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.26T + 37T^{2} \)
41 \( 1 + 7.59T + 41T^{2} \)
43 \( 1 + (3.97 + 6.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.10T + 47T^{2} \)
53 \( 1 + (3.75 + 6.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 0.332T + 61T^{2} \)
67 \( 1 + (6.45 - 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.81 - 3.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.48 - 4.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.58 + 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.30 - 10.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.569 + 0.985i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.87 - 3.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.37142948548621868456907643775, −9.760192933287520117903714117303, −9.347864639435502578547184744500, −8.595313884127861712480743513181, −6.96619654774853193311367524598, −5.92354376456738581421051931988, −5.14421550311403947541582720934, −3.11953769568289282904829948267, −2.41322069964859690676396798062, −1.22434994590789118317385350211, 1.39399469628796473409627902113, 3.10064468967674010048719971477, 4.98763092139548359475236061495, 5.97393159164869331255346958295, 6.42576027574142485116794238063, 7.43430564293620528440635105453, 8.384725199390563356424398131879, 9.345312570656801346252320107601, 10.10677091433724775553679083324, 10.34589922361292024883201789967

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