Properties

Label 2-2592-8.5-c1-0-20
Degree $2$
Conductor $2592$
Sign $0.707 + 0.707i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
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  − 2i·5-s − 4·7-s + 3i·11-s + 2i·13-s + 5·17-s − i·19-s − 2·23-s + 25-s + 4·31-s + 8i·35-s + 2i·37-s − 5·41-s − 11i·43-s + 6·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.894i·5-s − 1.51·7-s + 0.904i·11-s + 0.554i·13-s + 1.21·17-s − 0.229i·19-s − 0.417·23-s + 0.200·25-s + 0.718·31-s + 1.35i·35-s + 0.328i·37-s − 0.780·41-s − 1.67i·43-s + 0.875·47-s + 1.28·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.344405174\)
\(L(\frac12)\) \(\approx\) \(1.344405174\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−8.877039115066084812017234753564, −8.114126714555162090223280194443, −7.12344649649113331973845811093, −6.58081543741336949721027823851, −5.64813379326229438411499386793, −4.87014942433607626347001482681, −3.95713403100406123580054285921, −3.12776852230276948505832194140, −1.94371899044137482646927338356, −0.61110199134727875348818370995, 0.855496367201872642241971048822, 2.65146298955458765661610940902, 3.20857546817459366848380908624, 3.83937629775882937090934648874, 5.27899531674642223181679936990, 6.11434039723828817737245651052, 6.49707332372177201036679863871, 7.42835528083172887580093127716, 8.130698351421601421988342837329, 9.056790039904046581443592568504

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