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Properties

Label 2-5292-63.59-c1-0-39
Degree $2$
Conductor $5292$
Sign $-0.767 + 0.641i$
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  + 2.86·5-s − 2.71i·11-s + (−3.18 − 1.84i)13-s + (−3.22 + 5.58i)17-s + (−2.73 + 1.58i)19-s − 2.99i·23-s + 3.22·25-s + (2.48 − 1.43i)29-s + (−8.26 + 4.77i)31-s + (−1.70 − 2.95i)37-s + (0.794 − 1.37i)41-s + (−4.67 − 8.10i)43-s + (5.65 − 9.79i)47-s + (2.16 + 1.24i)53-s − 7.78i·55-s + ⋯
L(s)  = 1  + 1.28·5-s − 0.817i·11-s + (−0.884 − 0.510i)13-s + (−0.781 + 1.35i)17-s + (−0.628 + 0.362i)19-s − 0.623i·23-s + 0.645·25-s + (0.461 − 0.266i)29-s + (−1.48 + 0.857i)31-s + (−0.280 − 0.485i)37-s + (0.124 − 0.214i)41-s + (−0.713 − 1.23i)43-s + (0.824 − 1.42i)47-s + (0.297 + 0.171i)53-s − 1.04i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.767 + 0.641i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.767 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9199238277\)
\(L(\frac12)\) \(\approx\) \(0.9199238277\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 + (3.18 + 1.84i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 - 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.99iT - 23T^{2} \)
29 \( 1 + (-2.48 + 1.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.794 + 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.67 + 8.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.16 - 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 + 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.566 - 0.327i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 + 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (11.0 + 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.92 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.2 - 7.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.062289912121526516453181904700, −7.02691891251225238865965148633, −6.41690301943313768417852230230, −5.70791571598071592486999245424, −5.24714192783035084248801726860, −4.19429470746029233507258364801, −3.32794984650333206005668808411, −2.25888101075646879395961483019, −1.74269433855563733566514808138, −0.20925499288655817802390033562, 1.46130727339361145125848451866, 2.28509092101942960136409075097, 2.85078865600273037310387960850, 4.29925352038283687111960451229, 4.80025567824136717675806893625, 5.58796438284922990304419149191, 6.31857590653028326075799846653, 7.11901822764564220041640938232, 7.46329185536976910988504368709, 8.689353502215801893036059491313

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