Properties

Label 2-5292-63.59-c1-0-13
Degree $2$
Conductor $5292$
Sign $-0.0373 - 0.999i$
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  + 2.18·5-s − 1.46i·11-s + (2.92 + 1.69i)13-s + (−1.32 + 2.28i)17-s + (−6.87 + 3.97i)19-s + 4.00i·23-s − 0.234·25-s + (6.71 − 3.87i)29-s + (−0.612 + 0.353i)31-s + (1.41 + 2.45i)37-s + (−3.74 + 6.48i)41-s + (−1.27 − 2.20i)43-s + (−6.27 + 10.8i)47-s + (−2.41 − 1.39i)53-s − 3.19i·55-s + ⋯
L(s)  = 1  + 0.976·5-s − 0.441i·11-s + (0.811 + 0.468i)13-s + (−0.320 + 0.555i)17-s + (−1.57 + 0.911i)19-s + 0.836i·23-s − 0.0468·25-s + (1.24 − 0.719i)29-s + (−0.109 + 0.0634i)31-s + (0.233 + 0.403i)37-s + (−0.584 + 1.01i)41-s + (−0.193 − 0.335i)43-s + (−0.915 + 1.58i)47-s + (−0.331 − 0.191i)53-s − 0.431i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0373 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0373 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0373 - 0.999i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.0373 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.787441094\)
\(L(\frac12)\) \(\approx\) \(1.787441094\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18T + 5T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 + (-2.92 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.32 - 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.87 - 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.00iT - 23T^{2} \)
29 \( 1 + (-6.71 + 3.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.612 - 0.353i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.41 - 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.27 - 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.41 + 1.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.71 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.75 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.92 + 5.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (-3.95 - 2.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.69 - 8.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.70 + 2.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.61 + 8.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.38 + 3.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.461811578459060773233675850104, −7.82213270379053237560389137827, −6.63998278176402425478524883451, −6.18108858966390814524016258893, −5.77153190615196144236669138439, −4.64606173428610959716144775087, −3.98051083217600829794409967112, −3.01670677966671048104082787229, −1.99221414454154674086713240566, −1.32060068802781791945762700278, 0.44398279700411321258133651664, 1.79446559018726628636417073854, 2.44048534559053793447194651362, 3.41253759096515964584081083312, 4.48409268825665204117508817640, 5.04345079245050859208478811618, 5.95580195770095983428277164343, 6.57015114790163927048536617473, 7.05941878285472939975361856711, 8.233907468290579542144543904636

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