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Properties

Label 2-5292-63.20-c1-0-26
Degree $2$
Conductor $5292$
Sign $0.0892 + 0.996i$
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 − 2.64·17-s − 7.94i·19-s + (−3.47 + 2.00i)23-s + (0.117 − 0.203i)25-s + (6.71 + 3.87i)29-s + (−0.612 + 0.353i)31-s − 2.83·37-s + (3.74 + 6.48i)41-s + (−1.27 + 2.20i)43-s + (6.27 − 10.8i)47-s − 2.79i·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 − 0.640·17-s − 1.82i·19-s + (−0.724 + 0.418i)23-s + (0.0234 − 0.0406i)25-s + (1.24 + 0.719i)29-s + (−0.109 + 0.0634i)31-s − 0.466·37-s + (0.584 + 1.01i)41-s + (−0.193 + 0.335i)43-s + (0.915 − 1.58i)47-s − 0.383i·53-s − 0.431i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0892 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0892 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.0892 + 0.996i$
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: \((2,\ 5292,\ (\ :1/2),\ 0.0892 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.058491663\)
\(L(\frac12)\) \(\approx\) \(1.058491663\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 7.94iT - 19T^{2} \)
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 \( 1 + (0.612 - 0.353i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
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 \( 1 + 2.79iT - 53T^{2} \)
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 \( 1 + 4.57iT - 73T^{2} \)
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 \( 1 + 9.23T + 89T^{2} \)
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

−7.990069624663294792234385246211, −7.06890193777009746133946240994, −6.73149053863278128510148196825, −6.00985026387510401696279485627, −4.97325620391552827871207418630, −4.53517920397229547006689746227, −3.25825755846765986186960813006, −2.63759599428971414882131146337, −1.87840521223247248567687647501, −0.28278913711705324898742303348, 1.09873227209878026678647526472, 2.09237370449677826939900885263, 2.89540738788258444367768527452, 4.16375519481756609709973837729, 4.60461479783560317518217414598, 5.71745537168267903242802439347, 5.83566987062227561675871925127, 7.02279077194345354494517816070, 7.73376870523802023484143299913, 8.357738490543125724621116650513

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