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Properties

Label 2-5292-63.38-c1-0-17
Degree $2$
Conductor $5292$
Sign $0.936 - 0.351i$
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  + (1.09 − 1.89i)5-s + (1.26 − 0.732i)11-s + (−2.92 + 1.69i)13-s + (1.32 − 2.28i)17-s + (−6.87 + 3.97i)19-s + (3.47 + 2.00i)23-s + (0.117 + 0.203i)25-s + (6.71 + 3.87i)29-s − 0.706i·31-s + (1.41 + 2.45i)37-s + (3.74 + 6.48i)41-s + (−1.27 + 2.20i)43-s − 12.5·47-s + (2.41 + 1.39i)53-s − 3.19i·55-s + ⋯
L(s)  = 1  + (0.488 − 0.845i)5-s + (0.382 − 0.220i)11-s + (−0.811 + 0.468i)13-s + (0.320 − 0.555i)17-s + (−1.57 + 0.911i)19-s + (0.724 + 0.418i)23-s + (0.0234 + 0.0406i)25-s + (1.24 + 0.719i)29-s − 0.126i·31-s + (0.233 + 0.403i)37-s + (0.584 + 1.01i)41-s + (−0.193 + 0.335i)43-s − 1.83·47-s + (0.331 + 0.191i)53-s − 0.431i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.936 - 0.351i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.936 - 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.938921821\)
\(L(\frac12)\) \(\approx\) \(1.938921821\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.26 + 0.732i)T + (5.5 - 9.52i)T^{2} \)
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 \( 1 + (-3.47 - 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.71 - 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.706iT - 31T^{2} \)
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 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + (-2.41 - 1.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 7.79iT - 61T^{2} \)
67 \( 1 - 5.84T + 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-3.95 - 2.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.38T + 79T^{2} \)
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

−8.396925769922737040442711274766, −7.56773036966679753821087394716, −6.66087534138761291261095715291, −6.17749087728465261007300205534, −5.10690134525080975236613597940, −4.78888904219985587308661975946, −3.83082541778144476282320849791, −2.82040439362102368775995790828, −1.83402895424810840473668201931, −0.953669073226929319944918935376, 0.60111878161945676788049361807, 2.14257413759579223536201930076, 2.58034553377460396958804744292, 3.59966338485815096724859064909, 4.52066852839143921336333349098, 5.20733586192072331126955619130, 6.24830787397294541183664514114, 6.64218820381795970298122900017, 7.26131195675157758014962896513, 8.251428302099732506425968455541

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