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Properties

Label 2-2592-72.59-c1-0-18
Degree $2$
Conductor $2592$
Sign $0.766 - 0.642i$
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  + (2.44 − 1.41i)11-s + 5.65i·17-s − 2·19-s + (2.5 + 4.33i)25-s + (9.79 + 5.65i)41-s + (−5 − 8.66i)43-s + (−3.5 + 6.06i)49-s + (12.2 + 7.07i)59-s + (7 − 12.1i)67-s + 2·73-s + (2.44 − 1.41i)83-s + 5.65i·89-s + (5 + 8.66i)97-s + 19.7i·107-s + (9.79 + 5.65i)113-s + ⋯
L(s)  = 1  + (0.738 − 0.426i)11-s + 1.37i·17-s − 0.458·19-s + (0.5 + 0.866i)25-s + (1.53 + 0.883i)41-s + (−0.762 − 1.32i)43-s + (−0.5 + 0.866i)49-s + (1.59 + 0.920i)59-s + (0.855 − 1.48i)67-s + 0.234·73-s + (0.268 − 0.155i)83-s + 0.599i·89-s + (0.507 + 0.879i)97-s + 1.91i·107-s + (0.921 + 0.532i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771125924\)
\(L(\frac12)\) \(\approx\) \(1.771125924\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.44 + 1.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-9.79 - 5.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-12.2 - 7.07i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.44 + 1.41i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.905663152164363441061381210338, −8.302959890026293213760211698034, −7.47244811616173794816824830077, −6.54280943500618081236617962249, −6.01480826638229703175892113185, −5.06372646575781099861447918720, −4.05785889180958603539574212928, −3.41835976587424883770461151179, −2.17609041993992748506461114324, −1.08159481670629299401476962885, 0.69253233935336052018524705279, 2.04712106502501599294240170019, 2.99655211863693513806503826089, 4.09227506931268718778456219576, 4.77895727967667629747313549830, 5.68802807054375817658526257788, 6.67224820975402890367874832112, 7.10862492926473859187945820194, 8.103819355268825834510514312512, 8.801864744275687326411568157761

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