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Properties

Label 2-5292-63.41-c1-0-25
Degree $2$
Conductor $5292$
Sign $0.973 + 0.230i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604113323\)
\(L(\frac12)\) \(\approx\) \(1.604113323\)
\(L(\frac{3}{2})\) 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

−8.285282304290006865573314432946, −7.29772646412595127767261672895, −6.80013961447960398401697349996, −6.22147024298188294336707101319, −5.16591963446161733161743100300, −4.48770799196635542811787011308, −3.54901487267827837646120663510, −2.90949976603910217031410514876, −1.97445979581327926318763712097, −0.56527144012727182582113018823, 0.857327112105995400551063370002, 1.70043872757109033960191384821, 3.07376948597287621662723959333, 3.73325738638284390100544391339, 4.50647069269930485912894519329, 5.42024642269507655994877845858, 5.88020686721907586674498950213, 6.77354038410287113748426855299, 7.85161056988190829242649985009, 8.152222400974352600808634367127

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