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Properties

Label 2-2592-9.4-c1-0-10
Degree $2$
Conductor $2592$
Sign $-0.939 - 0.342i$
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  + (−1.80 + 3.12i)5-s + (1.80 + 3.12i)7-s + (0.5 + 0.866i)11-s + (−2 + 3.46i)13-s + 7.21·19-s + (−3 + 5.19i)23-s + (−4 − 6.92i)25-s + (3.60 + 6.24i)29-s + (−1.80 + 3.12i)31-s − 13·35-s + 10·37-s + (3.60 − 6.24i)41-s + (3.60 + 6.24i)43-s + (−5 − 8.66i)47-s + (−3 + 5.19i)49-s + ⋯
L(s)  = 1  + (−0.806 + 1.39i)5-s + (0.681 + 1.18i)7-s + (0.150 + 0.261i)11-s + (−0.554 + 0.960i)13-s + 1.65·19-s + (−0.625 + 1.08i)23-s + (−0.800 − 1.38i)25-s + (0.669 + 1.15i)29-s + (−0.323 + 0.560i)31-s − 2.19·35-s + 1.64·37-s + (0.563 − 0.975i)41-s + (0.549 + 0.952i)43-s + (−0.729 − 1.26i)47-s + (−0.428 + 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.473423540\)
\(L(\frac12)\) \(\approx\) \(1.473423540\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 3.12i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.60 - 6.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.80 - 3.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-3.60 + 6.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.60 - 6.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.60 + 6.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + (7.21 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.21T + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.293277571425363621432439178556, −8.365152070618854806088073167231, −7.49476223959021585609593711564, −7.17222106132729672211939426518, −6.19271705233552617084755607068, −5.34622796507958499147967426166, −4.46381645839758394740183988380, −3.41070504448785991658321012804, −2.70462959452620193399249822427, −1.68318283003199357496676854302, 0.56782978560851440509180456306, 1.14777182141211587461170941923, 2.79205438745246358812438743465, 4.02330002643322595527551882307, 4.47439505272143998499153413726, 5.20280944208865481678216282791, 6.12496418276968522051335588216, 7.41251120099253123659303242832, 7.908192580918260836650755768472, 8.203514452227344003003161061677

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