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Properties

Label 2-2592-4.3-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9995460660\)
\(L(\frac12)\) \(\approx\) \(0.9995460660\)
\(L(1)\) 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

−8.835329562967926538424880837699, −8.112288643434598400253925395328, −7.46489533649486724979761608553, −6.85179677358141361737839254646, −6.00839659131286349590281514734, −4.75780852264955011725612552843, −4.14383181013846604157199045310, −3.54995956504599165764533916323, −2.22644111320541678763386037923, −0.78718758488641036363558321646, 1.20711641630082962708312051515, 2.72199718680574180494921541263, 3.47206565787572430310233681330, 4.28593766155107304276178759891, 5.40216932836026774905950025996, 5.99826316913366299135280166196, 6.84583072068990107345937316356, 7.88143495530574624462483067782, 8.383884194867062998282570392503, 8.941211478666527159967055396356

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