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Properties

Label 2-2592-9.4-c1-0-11
Degree $2$
Conductor $2592$
Sign $-0.173 - 0.984i$
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  + (−1 + 1.73i)5-s + (0.5 + 0.866i)7-s + (1 + 1.73i)11-s + (−0.5 + 0.866i)13-s + 6·17-s + 5·19-s + (−3 + 5.19i)23-s + (0.500 + 0.866i)25-s + (−4 − 6.92i)29-s + (4 − 6.92i)31-s − 1.99·35-s − 5·37-s + (−4 + 6.92i)41-s + (−2 − 3.46i)43-s + (5 + 8.66i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.188 + 0.327i)7-s + (0.301 + 0.522i)11-s + (−0.138 + 0.240i)13-s + 1.45·17-s + 1.14·19-s + (−0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + (−0.742 − 1.28i)29-s + (0.718 − 1.24i)31-s − 0.338·35-s − 0.821·37-s + (−0.624 + 1.08i)41-s + (−0.304 − 0.528i)43-s + (0.729 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586767396\)
\(L(\frac12)\) \(\approx\) \(1.586767396\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (4 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.305162150103503589904365861545, −8.048628667283817133572531591472, −7.61116125768748048122857742517, −6.97978432223247837719173769582, −5.91818315949575802054117332782, −5.34276472833932407969641190144, −4.16450389275578763840297775193, −3.44522338230579655522703000602, −2.51818703227741552165695300142, −1.30170355327052187319400668430, 0.58634323116970206231022730702, 1.54959302163224183759014172545, 3.12937001845404679352790588891, 3.74100327052668068585441883213, 4.88468629897206806418541197235, 5.32980517024360692403866744061, 6.36833677694888569057432515143, 7.30707336722009469309896945365, 7.943505167343519544669533307697, 8.651012364970781627165330352333

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