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Properties

Label 2-2592-216.67-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.286 - 0.957i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 + 1.20i)11-s + (0.173 − 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (−0.266 + 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (1.17 + 0.984i)59-s + (0.326 − 1.85i)67-s + (−0.173 − 0.300i)73-s + (−0.173 − 0.984i)83-s + (−0.5 − 0.866i)89-s + (−1.43 + 1.20i)97-s + 0.347·107-s + (0.766 + 0.642i)113-s + ⋯
L(s)  = 1  + (−1.43 + 1.20i)11-s + (0.173 − 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (−0.266 + 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (1.17 + 0.984i)59-s + (0.326 − 1.85i)67-s + (−0.173 − 0.300i)73-s + (−0.173 − 0.984i)83-s + (−0.5 − 0.866i)89-s + (−1.43 + 1.20i)97-s + 0.347·107-s + (0.766 + 0.642i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.286 - 0.957i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1711, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9906048754\)
\(L(\frac12)\) \(\approx\) \(0.9906048754\)
\(L(1)\) 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 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.385194169726431946057527077303, −8.302894055649512105576760354994, −7.63382943529902681853348281556, −7.17943379888656645000945311835, −6.04201288044513282668208886836, −5.26259457165386812163072714609, −4.63668917642408023719454921272, −3.50150686272382424135787698450, −2.59891757200664342494343042608, −1.51146481085355359407134417322, 0.65642082288869425926984251620, 2.35623851475733170797214501753, 3.06809177748042540077109202365, 4.07969017726788857856444149683, 5.25369891368655717889394318999, 5.56917451837576997165904534345, 6.69487609997346532984228161211, 7.38345932621670333068301091112, 8.348889001260723865903804531257, 8.646761586536275837947463794328

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