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Properties

Label 2-5292-63.38-c1-0-34
Degree $2$
Conductor $5292$
Sign $-0.295 + 0.955i$
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  + (1.43 − 2.48i)5-s + (2.34 − 1.35i)11-s + (3.18 − 1.84i)13-s + (3.22 − 5.58i)17-s + (−2.73 + 1.58i)19-s + (−2.59 − 1.49i)23-s + (−1.61 − 2.79i)25-s + (2.48 + 1.43i)29-s − 9.54i·31-s + (−1.70 − 2.95i)37-s + (−0.794 − 1.37i)41-s + (−4.67 + 8.10i)43-s + 11.3·47-s + (−2.16 − 1.24i)53-s − 7.78i·55-s + ⋯
L(s)  = 1  + (0.641 − 1.11i)5-s + (0.708 − 0.408i)11-s + (0.884 − 0.510i)13-s + (0.781 − 1.35i)17-s + (−0.628 + 0.362i)19-s + (−0.540 − 0.311i)23-s + (−0.322 − 0.558i)25-s + (0.461 + 0.266i)29-s − 1.71i·31-s + (−0.280 − 0.485i)37-s + (−0.124 − 0.214i)41-s + (−0.713 + 1.23i)43-s + 1.64·47-s + (−0.297 − 0.171i)53-s − 1.04i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.295 + 0.955i$
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: \((2,\ 5292,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331243876\)
\(L(\frac12)\) \(\approx\) \(2.331243876\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 2.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 - 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.48 - 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.54iT - 31T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + (2.16 + 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.67T + 59T^{2} \)
61 \( 1 + 0.654iT - 61T^{2} \)
67 \( 1 - 7.72T + 67T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (11.0 + 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + (7.92 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.14 - 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.141218671496711031327273864565, −7.35300641707175342439298307483, −6.28451598740492783426355892075, −5.85320046098323583931011807987, −5.13827101316519364312157891124, −4.32518415202058235614008307164, −3.53835432425089183359846667446, −2.48678926142994276482157607055, −1.39363952627593428361606926497, −0.64161528420349713396607483210, 1.40395138314900814502393107286, 2.06735198994816135965619186560, 3.19423406081594637023945223026, 3.79319132715854420994204976763, 4.68257590739920306113976989559, 5.81420735750009099971981755325, 6.30049042858929274445537332585, 6.80976711815222051381278169212, 7.55768381668039273784498605944, 8.580424584226741453622969229736

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