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Properties

Label 2-5292-63.20-c1-0-15
Degree $2$
Conductor $5292$
Sign $0.261 - 0.965i$
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  + (1.37 + 2.37i)5-s + (−0.362 − 0.209i)11-s + (−1.32 + 0.765i)13-s + 3.90·17-s + 5.91i·19-s + (7.72 − 4.46i)23-s + (−1.26 + 2.18i)25-s + (−6.00 − 3.46i)29-s + (3.05 − 1.76i)31-s + 9.09·37-s + (1.06 + 1.84i)41-s + (−5.77 + 10.0i)43-s + (0.885 − 1.53i)47-s − 3.92i·53-s − 1.14i·55-s + ⋯
L(s)  = 1  + (0.613 + 1.06i)5-s + (−0.109 − 0.0630i)11-s + (−0.367 + 0.212i)13-s + 0.947·17-s + 1.35i·19-s + (1.61 − 0.930i)23-s + (−0.252 + 0.437i)25-s + (−1.11 − 0.643i)29-s + (0.548 − 0.316i)31-s + 1.49·37-s + (0.165 + 0.287i)41-s + (−0.881 + 1.52i)43-s + (0.129 − 0.223i)47-s − 0.538i·53-s − 0.154i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.261 - 0.965i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245396135\)
\(L(\frac12)\) \(\approx\) \(2.245396135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 - 2.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.362 + 0.209i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 - 5.91iT - 19T^{2} \)
23 \( 1 + (-7.72 + 4.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.05 + 1.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.09T + 37T^{2} \)
41 \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.885 + 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + (-2.02 - 3.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 - 0.932i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.38 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 - 1.90iT - 73T^{2} \)
79 \( 1 + (-0.433 + 0.751i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.45 + 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.77T + 89T^{2} \)
97 \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.129311166539401796896281983039, −7.67292159179468221878367160514, −6.80128306717317301003833595902, −6.24372489441956323555216313323, −5.60191046560931833615399962509, −4.72145351910467562454205012516, −3.74463410621365236203656714529, −2.91032009277007594180206736062, −2.27429496604556790689559022205, −1.09092298114631393510875168456, 0.67881500352154278443521027194, 1.52721040948125449463255261985, 2.62841450264198534738824523614, 3.47297629562975309052204424093, 4.57599254203866589884896303608, 5.30120837024172105120623792508, 5.50434580370348095332342095368, 6.70903951646890627323751227181, 7.31368404482803321839808229768, 8.084017788393135528981741003511

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