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Properties

Label 2-2592-9.7-c1-0-45
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 2.29i)5-s + (0.456 − 0.791i)7-s + (2.79 − 4.83i)11-s + (−1.29 − 2.23i)13-s − 1.73·17-s − 7.84·19-s + (−0.791 − 1.37i)23-s + (−1 + 1.73i)25-s + (0.409 − 0.708i)29-s + (0.913 + 1.58i)31-s − 2.41·35-s − 6.58·37-s + (4.37 + 7.58i)41-s + (−3.92 + 6.79i)43-s + (4 − 6.92i)47-s + ⋯
L(s)  = 1  + (−0.591 − 1.02i)5-s + (0.172 − 0.299i)7-s + (0.841 − 1.45i)11-s + (−0.358 − 0.620i)13-s − 0.420·17-s − 1.79·19-s + (−0.164 − 0.285i)23-s + (−0.200 + 0.346i)25-s + (0.0759 − 0.131i)29-s + (0.164 + 0.284i)31-s − 0.408·35-s − 1.08·37-s + (0.683 + 1.18i)41-s + (−0.597 + 1.03i)43-s + (0.583 − 1.01i)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: $\chi_{2592} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7460269058\)
\(L(\frac12)\) \(\approx\) \(0.7460269058\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.456 + 0.791i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.79 + 4.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.29 + 2.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 7.84T + 19T^{2} \)
23 \( 1 + (0.791 + 1.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.409 + 0.708i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.913 - 1.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + (-4.37 - 7.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.92 - 6.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.75T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.92 + 6.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.58T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (7.38 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.58 - 2.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.85T + 89T^{2} \)
97 \( 1 + (6.58 - 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.492537900551748346999505715748, −8.042342316953191092942258813414, −6.88693328087120163280398410735, −6.18259910958350910428955858270, −5.28967797358904314036481722760, −4.35926829322039109368341069422, −3.86006916984200532860640355997, −2.67297397328646269781661959864, −1.24926050375152829675627418786, −0.25431853627059158101449777506, 1.87024338168853367033371422859, 2.50633369507743075515538851017, 3.93828668358774840673288105321, 4.22720583338478015056296429718, 5.37926398170058651738988933427, 6.55657086406802226886045822753, 6.94068185547024931716739384555, 7.53256250555162069960035431811, 8.657569579771525929326933401762, 9.132165687002612071103344591639

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