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Properties

Label 2-72e2-12.11-c1-0-90
Degree $2$
Conductor $5184$
Sign $-i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46i·5-s − 3.46i·7-s − 3·11-s − 4·13-s + 1.73i·17-s + 1.73i·19-s − 6.99·25-s + 3.46i·29-s − 11.9·35-s − 2·37-s + 5.19i·41-s − 5.19i·43-s + 12·47-s − 4.99·49-s + 10.3i·55-s + ⋯
L(s)  = 1  − 1.54i·5-s − 1.30i·7-s − 0.904·11-s − 1.10·13-s + 0.420i·17-s + 0.397i·19-s − 1.39·25-s + 0.643i·29-s − 2.02·35-s − 0.328·37-s + 0.811i·41-s − 0.792i·43-s + 1.75·47-s − 0.714·49-s + 1.40i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (5183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + 5.19iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 15T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 3.46iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−7.62611704322947138936574020204, −7.27072892108541548380484851568, −6.12541428046831450641656521625, −5.29541209981851289314870824777, −4.69527357879641793342988873957, −4.17793356059083870042674719010, −3.16108781818622316317540530935, −1.92662980715842389527602360838, −0.985005037981796487380401561060, 0, 2.08465067806497984259645401232, 2.69308387970470332083767621438, 3.07815950505512556271912172336, 4.38116493385442532726123958805, 5.25729890067862831688816183483, 5.89844809057486288426593256482, 6.57704499976311611594225050833, 7.50441035974455279455752874791, 7.67350903779401724855999508825

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