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Properties

Label 2-5292-63.47-c1-0-35
Degree $2$
Conductor $5292$
Sign $-0.950 + 0.310i$
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  − 0.184·5-s − 2.14i·11-s + (−0.136 + 0.0789i)13-s + (1.22 + 2.12i)17-s + (2.60 + 1.50i)19-s − 3.36i·23-s − 4.96·25-s + (−4.31 − 2.49i)29-s + (−2.50 − 1.44i)31-s + (−3.57 + 6.19i)37-s + (−5.13 − 8.89i)41-s + (−2.64 + 4.57i)43-s + (−1.45 − 2.51i)47-s + (1.08 − 0.626i)53-s + 0.395i·55-s + ⋯
L(s)  = 1  − 0.0823·5-s − 0.647i·11-s + (−0.0379 + 0.0219i)13-s + (0.297 + 0.514i)17-s + (0.598 + 0.345i)19-s − 0.701i·23-s − 0.993·25-s + (−0.800 − 0.462i)29-s + (−0.449 − 0.259i)31-s + (−0.588 + 1.01i)37-s + (−0.801 − 1.38i)41-s + (−0.402 + 0.697i)43-s + (−0.212 − 0.367i)47-s + (0.149 − 0.0860i)53-s + 0.0533i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.950 + 0.310i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3913903623\)
\(L(\frac12)\) \(\approx\) \(0.3913903623\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.184T + 5T^{2} \)
11 \( 1 + 2.14iT - 11T^{2} \)
13 \( 1 + (0.136 - 0.0789i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.60 - 1.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (4.31 + 2.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.50 + 1.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 - 6.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.13 + 8.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.64 - 4.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.45 + 2.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.08 + 0.626i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.41 - 2.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.96 + 4.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.50 - 2.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.336iT - 71T^{2} \)
73 \( 1 + (-6.56 + 3.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.75 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.35 + 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.57 - 9.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.29 + 3.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.051324943262199124742719854178, −7.15310906542029904068815710197, −6.42541733185947642388734731177, −5.66328648535468822176782831676, −5.09339244494485469195980065410, −3.94557157479388201103249025355, −3.49503954030429121983258114877, −2.39855429209170426334504502443, −1.41950364543507187894661318157, −0.10182354813269904335316590989, 1.35948951977915885620516155518, 2.27554112023100593793039876241, 3.33251076567060634219533121632, 3.97040633209417852221368210144, 5.06060739559856019869700507866, 5.42516940277268798063941271536, 6.44875245029188941168764934992, 7.22596451820907660072551671941, 7.64626400535737230974534551081, 8.468073425535742976975157439123

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