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Properties

Label 2-5292-63.41-c1-0-22
Degree $2$
Conductor $5292$
Sign $0.987 + 0.155i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.987 + 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.988699909\)
\(L(\frac12)\) \(\approx\) \(1.988699909\)
\(L(\frac{3}{2})\) 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

−8.136250138178157218347860656586, −7.59813457772473538635574229254, −6.64083407486485645019496461201, −6.08992699981090354724412044972, −5.15456666228062323237608538056, −4.66607096671974599971217504086, −3.60372962592256435779386929230, −2.87500964942529364961194053598, −1.80597604069889129967525237925, −0.76853482340162720527581499681, 0.77138553503408563290471651565, 2.01107819840466290713020191480, 2.78316208072812562929055134682, 3.79094050670480915511183728042, 4.41947097177823505233711005362, 5.54052828017905333090900513151, 5.87334434449874663003059972028, 6.96544296651522439297149131367, 7.33135881079103024196452189507, 8.290325838779232292192505771476

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