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Properties

Label 2-2592-4.3-c0-0-1
Degree $2$
Conductor $2592$
Sign $0.707 - 0.707i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + i·7-s + i·11-s + 13-s − i·23-s − 29-s + i·31-s + i·35-s − 41-s + i·43-s − i·47-s + i·55-s + i·59-s + 61-s + 65-s + ⋯
L(s)  = 1  + 5-s + i·7-s + i·11-s + 13-s − i·23-s − 29-s + i·31-s + i·35-s − 41-s + i·43-s − i·47-s + i·55-s + i·59-s + 61-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475326263\)
\(L(\frac12)\) \(\approx\) \(1.475326263\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.958360022064102644450629735256, −8.745118198834108793309262904072, −7.64272955309250517418948258730, −6.69213281627347974795697100029, −6.05491654107565598644557414359, −5.37948968053665317940103176965, −4.56257496683099762814728594362, −3.39368863719919148680598412521, −2.30844686996641907734699820501, −1.63685426782656834368027792148, 1.05524212247607180034315974689, 2.10901318746850904405151175653, 3.44695159936980779875178446219, 3.96895049333650488678674789935, 5.28711275844865197318019139107, 5.86588489901786147290552407167, 6.56787465966789100831738513382, 7.45593461183042937970334290333, 8.227913313471213013232453476468, 9.036113543597371675970571194912

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