Properties

Label 2-5292-63.58-c1-0-36
Degree $2$
Conductor $5292$
Sign $-0.841 + 0.540i$
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 − 1.73i)5-s + (2 − 3.46i)11-s + (1.5 − 2.59i)13-s + (−3.5 − 6.06i)17-s + (2.5 − 4.33i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (−0.5 − 0.866i)29-s + 3·31-s + (−5.5 + 9.52i)37-s + (4.5 − 7.79i)41-s + (−2.5 − 4.33i)43-s + 3·47-s + (1.5 + 2.59i)53-s − 7.99·55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.603 − 1.04i)11-s + (0.416 − 0.720i)13-s + (−0.848 − 1.47i)17-s + (0.573 − 0.993i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.0928 − 0.160i)29-s + 0.538·31-s + (−0.904 + 1.56i)37-s + (0.702 − 1.21i)41-s + (−0.381 − 0.660i)43-s + 0.437·47-s + (0.206 + 0.356i)53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.499188949\)
\(L(\frac12)\) \(\approx\) \(1.499188949\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
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 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
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 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
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

−8.044008778603206265035216160699, −7.13062751110142116872727820950, −6.59450139267222933618571231482, −5.53528593985476362032981996572, −5.04978739854270182573705953432, −4.22715439865998572172339636866, −3.34188629903753066842655143238, −2.62606682154743824723359755987, −1.12742308660158117373413333974, −0.45174014862334022527628124690, 1.41890205762233065849731444528, 2.20166741297343334647315815421, 3.34759211510826943030556262876, 4.02646712820557168961653096035, 4.60093741361838950831408494071, 5.75310812878798172400534994717, 6.54803649366268306762141582378, 6.91262350179843331823121520073, 7.72968583124454813501522639041, 8.425227464756000798854648875019

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