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Properties

Label 2-2592-9.5-c0-0-4
Degree $2$
Conductor $2592$
Sign $0.819 + 0.573i$
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  + (0.448 + 0.258i)5-s + (0.866 − 1.5i)13-s − 1.93i·17-s + (−0.366 − 0.633i)25-s + (−1.67 + 0.965i)29-s + 37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)49-s + 1.41i·53-s + (0.5 + 0.866i)61-s + (0.776 − 0.448i)65-s + 1.73·73-s + (0.499 − 0.866i)85-s − 0.517i·89-s + (−1.22 + 0.707i)101-s + ⋯
L(s)  = 1  + (0.448 + 0.258i)5-s + (0.866 − 1.5i)13-s − 1.93i·17-s + (−0.366 − 0.633i)25-s + (−1.67 + 0.965i)29-s + 37-s + (1.22 + 0.707i)41-s + (0.5 − 0.866i)49-s + 1.41i·53-s + (0.5 + 0.866i)61-s + (0.776 − 0.448i)65-s + 1.73·73-s + (0.499 − 0.866i)85-s − 0.517i·89-s + (−1.22 + 0.707i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.819 + 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.323275672\)
\(L(\frac12)\) \(\approx\) \(1.323275672\)
\(L(1)\) 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 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.93iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + 0.517iT - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.173535575881785897786903660623, −8.123666949418544241499554436077, −7.53098470584159558638377819745, −6.70733588585617104384980264450, −5.74709627100314517748282562868, −5.32655239942626429691933872333, −4.17717069620946883856514485815, −3.14410155713432365767001056232, −2.43771726693507515393862933515, −0.948367824448422089836272258093, 1.50508481208197083349854911807, 2.20240455667566436624183582313, 3.84211248126878035571129925807, 4.07653774983555271886413719405, 5.41995431055724860221592878372, 6.06726830731828025646944454993, 6.67734404244456466002034529203, 7.74595891764707767042332828669, 8.387700045909132893324434185972, 9.330984800832234277718495984077

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