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Properties

Label 2-5292-63.59-c1-0-36
Degree $2$
Conductor $5292$
Sign $0.381 + 0.924i$
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  + 2.74·5-s − 0.418i·11-s + (1.32 + 0.765i)13-s + (1.95 − 3.38i)17-s + (5.11 − 2.95i)19-s − 8.92i·23-s + 2.52·25-s + (−6.00 + 3.46i)29-s + (3.05 − 1.76i)31-s + (−4.54 − 7.87i)37-s + (−1.06 + 1.84i)41-s + (−5.77 − 10.0i)43-s + (−0.885 + 1.53i)47-s + (−3.39 − 1.96i)53-s − 1.14i·55-s + ⋯
L(s)  = 1  + 1.22·5-s − 0.126i·11-s + (0.367 + 0.212i)13-s + (0.473 − 0.820i)17-s + (1.17 − 0.678i)19-s − 1.86i·23-s + 0.505·25-s + (−1.11 + 0.643i)29-s + (0.548 − 0.316i)31-s + (−0.747 − 1.29i)37-s + (−0.165 + 0.287i)41-s + (−0.881 − 1.52i)43-s + (−0.129 + 0.223i)47-s + (−0.466 − 0.269i)53-s − 0.154i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.435365785\)
\(L(\frac12)\) \(\approx\) \(2.435365785\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 5T^{2} \)
11 \( 1 + 0.418iT - 11T^{2} \)
13 \( 1 + (-1.32 - 0.765i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.92iT - 23T^{2} \)
29 \( 1 + (6.00 - 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.05 + 1.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.77 + 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.885 - 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.39 + 1.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.02 + 3.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 - 0.932i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.38 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (1.65 + 0.952i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.433 + 0.751i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.45 + 5.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.88 - 8.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.200 + 0.115i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.162437030345565183201214924613, −7.07647174263560781980486898014, −6.76966213852183094453831050495, −5.68632715401580491039186581796, −5.39229009059974322467038428345, −4.48169250928585000490136364462, −3.41516800310402287085486676152, −2.58391601601483796097518782630, −1.77042244095130722031059733335, −0.62744995927451152463634776536, 1.36557996869813031041430794911, 1.79151922299149055138227300242, 3.11649149055761083892243794756, 3.64646549558475624899564557162, 4.87864125148637441524185629899, 5.59481518907999846845175061385, 5.99125437919631039914483441897, 6.79211274843156218736018647802, 7.75620753215821702669642047811, 8.171735814279762694212716809599

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