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Properties

Label 2-2592-648.355-c0-0-0
Degree $2$
Conductor $2592$
Sign $-0.627 + 0.778i$
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  + (−0.893 − 0.448i)3-s + (0.597 + 0.802i)9-s + (−1.14 + 0.754i)11-s + (−1.52 − 1.27i)17-s + (1.28 − 1.07i)19-s + (−0.993 − 0.116i)25-s + (−0.173 − 0.984i)27-s + (1.36 − 0.159i)33-s + (0.707 − 1.64i)41-s + (−0.707 − 0.355i)43-s + (0.893 − 0.448i)49-s + (0.786 + 1.82i)51-s + (−1.62 + 0.385i)57-s + (−0.0971 − 0.0639i)59-s + (−1.16 − 1.56i)67-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)3-s + (0.597 + 0.802i)9-s + (−1.14 + 0.754i)11-s + (−1.52 − 1.27i)17-s + (1.28 − 1.07i)19-s + (−0.993 − 0.116i)25-s + (−0.173 − 0.984i)27-s + (1.36 − 0.159i)33-s + (0.707 − 1.64i)41-s + (−0.707 − 0.355i)43-s + (0.893 − 0.448i)49-s + (0.786 + 1.82i)51-s + (−1.62 + 0.385i)57-s + (−0.0971 − 0.0639i)59-s + (−1.16 − 1.56i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.627 + 0.778i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ -0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4644989698\)
\(L(\frac12)\) \(\approx\) \(0.4644989698\)
\(L(1)\) 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 \( 1 + (0.893 + 0.448i)T \)
good5 \( 1 + (0.993 + 0.116i)T^{2} \)
7 \( 1 + (-0.893 + 0.448i)T^{2} \)
11 \( 1 + (1.14 - 0.754i)T + (0.396 - 0.918i)T^{2} \)
13 \( 1 + (-0.973 + 0.230i)T^{2} \)
17 \( 1 + (1.52 + 1.27i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (-1.28 + 1.07i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.893 - 0.448i)T^{2} \)
29 \( 1 + (0.286 + 0.957i)T^{2} \)
31 \( 1 + (0.835 - 0.549i)T^{2} \)
37 \( 1 + (0.939 + 0.342i)T^{2} \)
41 \( 1 + (-0.707 + 1.64i)T + (-0.686 - 0.727i)T^{2} \)
43 \( 1 + (0.707 + 0.355i)T + (0.597 + 0.802i)T^{2} \)
47 \( 1 + (0.835 + 0.549i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.0971 + 0.0639i)T + (0.396 + 0.918i)T^{2} \)
61 \( 1 + (0.0581 - 0.998i)T^{2} \)
67 \( 1 + (1.16 + 1.56i)T + (-0.286 + 0.957i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.12 + 0.408i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.686 - 0.727i)T^{2} \)
83 \( 1 + (0.137 + 0.318i)T + (-0.686 + 0.727i)T^{2} \)
89 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.0333 - 0.572i)T + (-0.993 + 0.116i)T^{2} \)
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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

−8.877407877195292858347346986660, −7.65334665323424459985370292225, −7.30626675521211230948325656689, −6.61955702082303582509488859860, −5.53723127603546974283140555434, −5.01965902958370734305429680098, −4.28958419633228078506930923968, −2.77136004461721299971778852809, −1.98953153326217870577205862833, −0.33807178280573322258089579352, 1.44726037270912485430207346901, 2.86838130599870498189137050137, 3.89535563086038904088759684667, 4.61648310399581539840866009523, 5.76160093337145137119985158987, 5.86702492971865821370435644702, 6.95814689079204190166707355753, 7.87277513064061502341782684798, 8.525354664692241667755084473081, 9.514259766792524515009455496388

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