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Properties

Label 2-2592-36.23-c1-0-31
Degree $2$
Conductor $2592$
Sign $0.996 + 0.0871i$
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)5-s + (1.73 − i)7-s + (1.41 + 2.44i)11-s + (1 − 1.73i)13-s − 6i·19-s + (2.82 − 4.89i)23-s + (1.49 + 2.59i)25-s + (2.44 − 1.41i)29-s + (−1.73 − i)31-s + 5.65·35-s + 6·37-s + (4.89 + 2.82i)41-s + (1.73 − i)43-s + (−5.65 − 9.79i)47-s + (−1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (1.09 + 0.632i)5-s + (0.654 − 0.377i)7-s + (0.426 + 0.738i)11-s + (0.277 − 0.480i)13-s − 1.37i·19-s + (0.589 − 1.02i)23-s + (0.299 + 0.519i)25-s + (0.454 − 0.262i)29-s + (−0.311 − 0.179i)31-s + 0.956·35-s + 0.986·37-s + (0.765 + 0.441i)41-s + (0.264 − 0.152i)43-s + (−0.825 − 1.42i)47-s + (−0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.996 + 0.0871i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.996 + 0.0871i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623304082\)
\(L(\frac12)\) \(\approx\) \(2.623304082\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 - 1.41i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 + 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 + i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-4.89 - 2.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.48iT - 53T^{2} \)
59 \( 1 + (-1.41 + 2.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (12.1 - 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.41 + 2.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
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

−8.977638119308830661793640272292, −8.099896171482015956918353637491, −7.18988101111267382134538835572, −6.62320029547921337943244733610, −5.86293888016994454338946965780, −4.89994935439988336629913157754, −4.23853831035387735416815991789, −2.88747660692505604855823144083, −2.20181157120466646990534716546, −1.01669137576574921274726427430, 1.25136627720119561209121655697, 1.85224502674862624640689973841, 3.13367173471261597716914016206, 4.18361259868437420391159204238, 5.14354325161090785848530249967, 5.78915690209916898095580174656, 6.31385517296824839511752567285, 7.46328223592148096032686895848, 8.312977616317509819388644772939, 8.912555243497991394337104083644

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