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Properties

Label 2-5292-63.20-c1-0-2
Degree $2$
Conductor $5292$
Sign $-0.294 - 0.955i$
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  + (−0.349 − 0.605i)5-s + (0.229 + 0.132i)11-s + (1.13 − 0.657i)13-s − 3.72·17-s − 0.441i·19-s + (−4.29 + 2.48i)23-s + (2.25 − 3.90i)25-s + (0.273 + 0.157i)29-s + (−4.85 + 2.80i)31-s + 0.702·37-s + (5.39 + 9.34i)41-s + (3.73 − 6.46i)43-s + (−3.50 + 6.06i)47-s + 9.83i·53-s − 0.185i·55-s + ⋯
L(s)  = 1  + (−0.156 − 0.270i)5-s + (0.0692 + 0.0399i)11-s + (0.315 − 0.182i)13-s − 0.904·17-s − 0.101i·19-s + (−0.896 + 0.517i)23-s + (0.451 − 0.781i)25-s + (0.0507 + 0.0292i)29-s + (−0.872 + 0.503i)31-s + 0.115·37-s + (0.842 + 1.45i)41-s + (0.569 − 0.985i)43-s + (−0.510 + 0.884i)47-s + 1.35i·53-s − 0.0250i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8570659781\)
\(L(\frac12)\) \(\approx\) \(0.8570659781\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.349 + 0.605i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.229 - 0.132i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.13 + 0.657i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.72T + 17T^{2} \)
19 \( 1 + 0.441iT - 19T^{2} \)
23 \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.273 - 0.157i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.85 - 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.702T + 37T^{2} \)
41 \( 1 + (-5.39 - 9.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.50 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.83iT - 53T^{2} \)
59 \( 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.89 + 2.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.97 - 5.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 7.69iT - 73T^{2} \)
79 \( 1 + (0.698 - 1.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.72 + 6.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (9.18 + 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.328419823888493165082662650655, −7.81098924667462160557116861665, −6.94589111647104476627161246068, −6.25899852508193792989561213708, −5.55976542555005553238729682230, −4.62947828024429568207234890716, −4.07607352779368679631288943854, −3.10234401279720911036225616929, −2.17370950830499542087816745538, −1.10330467521976215848156246864, 0.23812833573828397108895137382, 1.67048777960493427815750758768, 2.52095945589610122102896736865, 3.54759231677134077010970095330, 4.18911120127577210863701577463, 5.03566198123390034501314731792, 5.94160293286285378833500828068, 6.51129837964575077371947131675, 7.30975693062598967191955149196, 7.893500308601105164948506812340

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