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Properties

Label 2-5292-63.16-c1-0-13
Degree $2$
Conductor $5292$
Sign $0.921 - 0.388i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  − 1.69·5-s − 2.47·11-s + (0.388 + 0.673i)13-s + (−1.40 − 2.43i)17-s + (−2.49 + 4.31i)19-s − 0.712·23-s − 2.11·25-s + (2.25 − 3.90i)29-s + (2.54 − 4.41i)31-s + (3.43 − 5.95i)37-s + (2.93 + 5.08i)41-s + (2.32 − 4.03i)43-s + (−6.49 − 11.2i)47-s + (0.944 + 1.63i)53-s + 4.21·55-s + ⋯
L(s)  = 1  − 0.760·5-s − 0.746·11-s + (0.107 + 0.186i)13-s + (−0.340 − 0.590i)17-s + (−0.572 + 0.990i)19-s − 0.148·23-s − 0.422·25-s + (0.418 − 0.725i)29-s + (0.457 − 0.793i)31-s + (0.565 − 0.979i)37-s + (0.458 + 0.794i)41-s + (0.354 − 0.614i)43-s + (−0.947 − 1.64i)47-s + (0.129 + 0.224i)53-s + 0.567·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.134317549\)
\(L(\frac12)\) \(\approx\) \(1.134317549\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.69T + 5T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 + (-0.388 - 0.673i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.40 + 2.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.49 - 4.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.712T + 23T^{2} \)
29 \( 1 + (-2.25 + 3.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.43 + 5.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.49 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.944 - 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.14 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.15 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-2.49 - 4.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.82 - 8.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.32 + 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.175425701435325863790695574556, −7.61338711796399441670154830984, −6.93309358369226608629104623542, −6.00905554614264056227128251906, −5.42059764680480726174163807358, −4.28634497733306417604624164443, −4.01135535430565140135308045253, −2.84028303947185946783948635045, −2.07986519419862441889554057260, −0.63692878131089099289166070402, 0.48896453360980445148251997448, 1.85541992427958333989668675999, 2.89309932672918827223075850974, 3.58584903578842765031172751793, 4.62370748066578361198468518604, 4.97212452238900744468230757552, 6.18746281689369893478429966973, 6.62051517305138763120553976655, 7.65924395383833817112779920374, 8.026834937419006748867145252725

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