Properties

Label 2-72e2-8.5-c1-0-66
Degree $2$
Conductor $5184$
Sign $-0.258 + 0.965i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  − 1.73i·5-s + 0.990·7-s + 2.10i·11-s + 6.36i·13-s − 3.81·17-s − 2i·19-s − 7.10·23-s + 2.00·25-s − 8.34i·29-s − 2.15·31-s − 1.71i·35-s − 4.62i·37-s − 0.816·41-s − 2.28i·43-s + 6.78·47-s + ⋯
L(s)  = 1  − 0.774i·5-s + 0.374·7-s + 0.633i·11-s + 1.76i·13-s − 0.925·17-s − 0.458i·19-s − 1.48·23-s + 0.400·25-s − 1.54i·29-s − 0.387·31-s − 0.290i·35-s − 0.761i·37-s − 0.127·41-s − 0.348i·43-s + 0.989·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.177350486\)
\(L(\frac12)\) \(\approx\) \(1.177350486\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 - 0.990T + 7T^{2} \)
11 \( 1 - 2.10iT - 11T^{2} \)
13 \( 1 - 6.36iT - 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 7.10T + 23T^{2} \)
29 \( 1 + 8.34iT - 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 + 4.62iT - 37T^{2} \)
41 \( 1 + 0.816T + 41T^{2} \)
43 \( 1 + 2.28iT - 43T^{2} \)
47 \( 1 - 6.78T + 47T^{2} \)
53 \( 1 - 3.14iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 4.87iT - 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 9.08T + 79T^{2} \)
83 \( 1 + 4.28iT - 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.059895073306938191443689173923, −7.29856216715785597673251763832, −6.56007959004539825325440291955, −5.90890982290637611388364203114, −4.71141280566438095053196868996, −4.54846411215706935050594596287, −3.71683526946443654512682029612, −2.14810490877487650784198203056, −1.86071904570206749859106116509, −0.32644302790460736632170400537, 1.08636370589864071113434304085, 2.33095571310737458410114255027, 3.11265893063159503727005184498, 3.77871941717672402735588710486, 4.83970457065056837580887226763, 5.61946814842963705847592613029, 6.20803363647151716636923715554, 7.02538537327231094704935123022, 7.74731130441630813718747532637, 8.344675471820503857147045976752

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