↑

Properties

Label 2-5292-63.4-c1-0-37
Degree $2$
Conductor $5292$
Sign $-0.371 + 0.928i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.239·5-s + 5.12·11-s + (2.44 − 4.23i)13-s + (1.85 − 3.20i)17-s + (−1.83 − 3.16i)19-s − 7.42·23-s − 4.94·25-s + (1.73 + 3.00i)29-s + (0.358 + 0.621i)31-s + (−2.30 − 3.98i)37-s + (2.80 − 4.85i)41-s + (−6.24 − 10.8i)43-s + (2.16 − 3.75i)47-s + (−0.471 + 0.816i)53-s − 1.22·55-s + ⋯
L(s)  = 1  − 0.106·5-s + 1.54·11-s + (0.677 − 1.17i)13-s + (0.449 − 0.777i)17-s + (−0.419 − 0.727i)19-s − 1.54·23-s − 0.988·25-s + (0.321 + 0.557i)29-s + (0.0644 + 0.111i)31-s + (−0.378 − 0.655i)37-s + (0.437 − 0.757i)41-s + (−0.952 − 1.64i)43-s + (0.316 − 0.548i)47-s + (−0.0647 + 0.112i)53-s − 0.165·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.506593766\)
\(L(\frac12)\) \(\approx\) \(1.506593766\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.239T + 5T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 + (-2.44 + 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.85 + 3.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 + 3.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.42T + 23T^{2} \)
29 \( 1 + (-1.73 - 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.358 - 0.621i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.30 + 3.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.471 - 0.816i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.75 - 4.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.330 - 0.571i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + (-1.83 + 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.11 + 5.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.85 - 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.74 + 6.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.945986044720313035030765675651, −7.27175760661778264321093595195, −6.52090903769136558978987783111, −5.82737460565490006874741280350, −5.16230278811630939780052010981, −3.98777975653233517822357665810, −3.68482171503282083742080513819, −2.55652352747526705254068207282, −1.49558502680031541556242891620, −0.40060167921662194071853160009, 1.38628258311960928711102103834, 1.89492014085407569796128799838, 3.32962794296751094530872644662, 4.09992400235426595471585753710, 4.38319683753353369394750890445, 5.82462468579558754497580006404, 6.28553806878087634755713345442, 6.74355313188018348030142281349, 7.996147781073506740433114686554, 8.198806412538789072949059330507

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