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Properties

Label 2-2592-72.13-c1-0-8
Degree $2$
Conductor $2592$
Sign $0.906 - 0.422i$
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  + (−3.23 − 1.86i)5-s + (−0.366 − 0.633i)7-s + (−4.09 + 2.36i)11-s + (−2.13 − 1.23i)13-s − 3.73·17-s + 3.26i·19-s + (4.36 − 7.56i)23-s + (4.46 + 7.73i)25-s + (−4.5 + 2.59i)29-s + (1 − 1.73i)31-s + 2.73i·35-s + 5i·37-s + (−2.26 + 3.92i)41-s + (8.36 − 4.83i)43-s + (1.73 + 3i)47-s + ⋯
L(s)  = 1  + (−1.44 − 0.834i)5-s + (−0.138 − 0.239i)7-s + (−1.23 + 0.713i)11-s + (−0.591 − 0.341i)13-s − 0.905·17-s + 0.749i·19-s + (0.910 − 1.57i)23-s + (0.892 + 1.54i)25-s + (−0.835 + 0.482i)29-s + (0.179 − 0.311i)31-s + 0.461i·35-s + 0.821i·37-s + (−0.354 + 0.613i)41-s + (1.27 − 0.736i)43-s + (0.252 + 0.437i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.906 - 0.422i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6493362866\)
\(L(\frac12)\) \(\approx\) \(0.6493362866\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 + 1.86i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.366 + 0.633i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.13 + 1.23i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 - 3.26iT - 19T^{2} \)
23 \( 1 + (-4.36 + 7.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 + (2.26 - 3.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.36 + 4.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.928iT - 53T^{2} \)
59 \( 1 + (-7.26 - 4.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.79 - 4.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.09 + 2.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.66T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-2.63 - 4.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.1 + 7i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.695224977703199624047815579981, −8.200507525608738759717967143756, −7.44469447095679338129043963512, −6.97029967071046909595713918877, −5.66597926143285930294170202492, −4.69944068835344600994320693818, −4.42485038872002155908455430484, −3.30308329041394444165461313003, −2.26872601464439317032929646687, −0.65560566286579133030351953302, 0.35429115146978957963573755423, 2.36699045680888375322315118691, 3.09439792911219591089330610541, 3.90399000499255491286348398014, 4.84201474195858837567206248105, 5.69692089716511367365678360826, 6.75299546248311619799005159807, 7.45900808866195960902854001164, 7.79998774068367156387969149496, 8.818048348591536578466778969070

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