Properties

Label 2-72e2-12.11-c1-0-29
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.93i·5-s + 0.732i·7-s + 5.27·11-s − 4.46·13-s + 0.896i·17-s − 1.26i·19-s − 1.41·23-s + 1.26·25-s + 5.41i·29-s − 7.46i·31-s − 1.41·35-s + 7.73·37-s + 0.378i·41-s + 8.73i·43-s + 4.62·47-s + ⋯
L(s)  = 1  + 0.863i·5-s + 0.276i·7-s + 1.59·11-s − 1.23·13-s + 0.217i·17-s − 0.290i·19-s − 0.294·23-s + 0.253·25-s + 1.00i·29-s − 1.34i·31-s − 0.239·35-s + 1.27·37-s + 0.0591i·41-s + 1.33i·43-s + 0.674·47-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: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851755711\)
\(L(\frac12)\) \(\approx\) \(1.851755711\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93iT - 5T^{2} \)
7 \( 1 - 0.732iT - 7T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 - 0.896iT - 17T^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 5.41iT - 29T^{2} \)
31 \( 1 + 7.46iT - 31T^{2} \)
37 \( 1 - 7.73T + 37T^{2} \)
41 \( 1 - 0.378iT - 41T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 - 2.44iT - 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 16.5iT - 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−8.458083998120940733182674209801, −7.39048339771305551803818209658, −7.07874441027862387668036394403, −6.26340014434339151570791405746, −5.67923779620765944315451973860, −4.55190096230183176114191381510, −3.98778099845930005194275828136, −2.93967134361160102205877263644, −2.31481062896495885493343332638, −1.09782282782195024766945104906, 0.55986025436932480098129400539, 1.50792570367219109194890040996, 2.53277128872875887941440238385, 3.70718814568077481357556232566, 4.34872004815808183275814667035, 5.00416848597170371709268889033, 5.84121394880183988530244960421, 6.67216293961768059554585411510, 7.29271587156324727309768517486, 8.035702958110777658714545862602

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