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Properties

Label 2-5292-63.38-c1-0-31
Degree $2$
Conductor $5292$
Sign $0.620 + 0.783i$
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  + (−1.48 + 2.57i)5-s + (4.09 − 2.36i)11-s + (3.54 − 2.04i)13-s + (−0.835 + 1.44i)17-s + (4.25 − 2.45i)19-s + (−4.25 − 2.45i)23-s + (−1.91 − 3.30i)25-s + (−0.238 − 0.137i)29-s + 1.60i·31-s + (−1.69 − 2.93i)37-s + (−3.55 − 6.15i)41-s + (5.22 − 9.05i)43-s − 10.9·47-s + (−0.707 − 0.408i)53-s + 14.0i·55-s + ⋯
L(s)  = 1  + (−0.664 + 1.15i)5-s + (1.23 − 0.712i)11-s + (0.981 − 0.566i)13-s + (−0.202 + 0.350i)17-s + (0.975 − 0.563i)19-s + (−0.886 − 0.511i)23-s + (−0.382 − 0.661i)25-s + (−0.0442 − 0.0255i)29-s + 0.287i·31-s + (−0.278 − 0.483i)37-s + (−0.555 − 0.961i)41-s + (0.797 − 1.38i)43-s − 1.60·47-s + (−0.0971 − 0.0560i)53-s + 1.89i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.620 + 0.783i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.620 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.572072885\)
\(L(\frac12)\) \(\approx\) \(1.572072885\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 2.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.835 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.25 + 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.25 + 2.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.238 + 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.60iT - 31T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.55 + 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (0.707 + 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 7.20iT - 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (13.6 + 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (4.03 - 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.60 - 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.00 + 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.115755934582645964327271496169, −7.27277136857433556663777721857, −6.67846567141362593231143010219, −6.10927791522086943498438106235, −5.30011765961042358228736484410, −4.05407807208525349788617678724, −3.58374007715243742053177357410, −2.97190091062950452058188944894, −1.73139684338361237408258065947, −0.47164921621660950351727199903, 1.13155235150422696965067720946, 1.64383124950260564483747049732, 3.15986964283708393170316925955, 4.09885652934196833136079334907, 4.38110040006025614903211570357, 5.33006625617871189771113810712, 6.16193043420721432859605972147, 6.87785194663773559886546162841, 7.69531950599190000207953790030, 8.349308004253430213522583443267

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