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Properties

Label 2-2592-9.4-c1-0-9
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  + (−0.5 + 0.866i)5-s + (1 + 1.73i)7-s + (1 + 1.73i)11-s + (−0.5 + 0.866i)13-s + 3·17-s − 2·19-s + (3 − 5.19i)23-s + (2 + 3.46i)25-s + (−0.5 − 0.866i)29-s + (−4 + 6.92i)31-s − 1.99·35-s + 37-s + (1 − 1.73i)41-s + (5 + 8.66i)43-s + (2 + 3.46i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.377 + 0.654i)7-s + (0.301 + 0.522i)11-s + (−0.138 + 0.240i)13-s + 0.727·17-s − 0.458·19-s + (0.625 − 1.08i)23-s + (0.400 + 0.692i)25-s + (−0.0928 − 0.160i)29-s + (−0.718 + 1.24i)31-s − 0.338·35-s + 0.164·37-s + (0.156 − 0.270i)41-s + (0.762 + 1.32i)43-s + (0.291 + 0.505i)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

\(L(1)\) \(\approx\) \(1.572105013\)
\(L(\frac12)\) \(\approx\) \(1.572105013\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
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 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.074219517910107170123169685523, −8.374157244671197229888976275219, −7.52153506196647021994677964460, −6.86089720500156528603793421175, −6.05295895014871378751704729591, −5.12159663933340461181337972183, −4.42455926541881354890315333849, −3.33540894067428484951595805272, −2.46556638195925482025849912338, −1.35091943175233921764231740144, 0.55197987167882812077438211262, 1.63510093334386925736397163060, 3.00388209189794220041371257209, 3.90391421929595213886614638042, 4.64008079958881044681754947750, 5.56990365191537811955551884258, 6.27498215342652458839939271201, 7.52936688303693474193662169505, 7.63095723746092022357266551524, 8.774174635496412521819271841852

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