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Properties

Label 2-2592-36.11-c1-0-44
Degree $2$
Conductor $2592$
Sign $-0.984 - 0.173i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 − 0.258i)5-s + (−2.36 − 1.36i)7-s + (−0.189 + 0.328i)11-s + (1.23 + 2.13i)13-s + 3.34i·17-s − 4.73i·19-s + (−0.707 − 1.22i)23-s + (−2.36 + 4.09i)25-s + (−5.91 − 3.41i)29-s + (0.464 − 0.267i)31-s − 1.41·35-s − 4.26·37-s + (4.57 − 2.63i)41-s + (−4.56 − 2.63i)43-s + (4.76 − 8.24i)47-s + ⋯
L(s)  = 1  + (0.200 − 0.115i)5-s + (−0.894 − 0.516i)7-s + (−0.0571 + 0.0989i)11-s + (0.341 + 0.591i)13-s + 0.811i·17-s − 1.08i·19-s + (−0.147 − 0.255i)23-s + (−0.473 + 0.819i)25-s + (−1.09 − 0.634i)29-s + (0.0833 − 0.0481i)31-s − 0.239·35-s − 0.701·37-s + (0.713 − 0.412i)41-s + (−0.695 − 0.401i)43-s + (0.694 − 1.20i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.448 + 0.258i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.189 - 0.328i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.34iT - 17T^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.91 + 3.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.464 + 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.26T + 37T^{2} \)
41 \( 1 + (-4.57 + 2.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.56 + 2.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.76 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.44iT - 53T^{2} \)
59 \( 1 + (-6.83 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.59 - 7.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.63 - 5.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.1T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (12.6 + 7.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.378 - 0.656i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.20iT - 89T^{2} \)
97 \( 1 + (7.19 - 12.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.817527326129370981337307337787, −7.48005574235438953574658177616, −7.05498756896310640828254066527, −6.13878962693140058265589719440, −5.51417993606161408641306706497, −4.29715757159106781314908688424, −3.73730981570104614282585370647, −2.63932492887798242713494506629, −1.49917519135787621217683654053, 0, 1.62434382095010210384324492643, 2.84004280653054999106605338497, 3.45293715400910425331042528911, 4.55120432158553745777056254034, 5.71029158406070230709079070144, 5.99104880076370483855852876151, 6.97321591310751979798450060764, 7.78356254590200289739133293553, 8.541191212346546610736504717384

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