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Properties

Label 2-5292-63.16-c1-0-6
Degree $2$
Conductor $5292$
Sign $-0.678 - 0.734i$
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  + 2·5-s − 4·11-s + (1.5 + 2.59i)13-s + (−3.5 − 6.06i)17-s + (2.5 − 4.33i)19-s − 4·23-s − 25-s + (−0.5 + 0.866i)29-s + (−1.5 + 2.59i)31-s + (−5.5 + 9.52i)37-s + (4.5 + 7.79i)41-s + (−2.5 + 4.33i)43-s + (−1.5 − 2.59i)47-s + (1.5 + 2.59i)53-s − 8·55-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s + (0.416 + 0.720i)13-s + (−0.848 − 1.47i)17-s + (0.573 − 0.993i)19-s − 0.834·23-s − 0.200·25-s + (−0.0928 + 0.160i)29-s + (−0.269 + 0.466i)31-s + (−0.904 + 1.56i)37-s + (0.702 + 1.21i)41-s + (−0.381 + 0.660i)43-s + (−0.218 − 0.378i)47-s + (0.206 + 0.356i)53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.678 - 0.734i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7495944749\)
\(L(\frac12)\) \(\approx\) \(0.7495944749\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.471569299830570261885680814790, −7.76256697761775094281284637422, −6.83801501672938365496937015119, −6.48005740464511607959397181931, −5.32699039620820098158153685314, −5.07883051308105902858202243707, −4.09738265007982544380199274375, −2.86701839339111701781496624199, −2.39540481917874066439325480381, −1.28688065587659384193810300401, 0.18111829181898944349505268921, 1.77699553025560590127593494969, 2.24866142287559010084300251927, 3.45223248475053897592685082813, 4.10773982465257530835407954156, 5.30912497191537879810876967483, 5.76244415660415257037642647171, 6.22261409301363136109878713499, 7.34737295339234893894023166121, 7.951161270402134281175056581958

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