↑

Properties

Label 2-2592-9.7-c1-0-28
Degree $2$
Conductor $2592$
Sign $-0.173 + 0.984i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−2.44 + 4.24i)7-s + (−2.44 + 4.24i)11-s + (−1.5 − 2.59i)13-s + 5·17-s − 4.89·19-s + (−2.44 − 4.24i)23-s + (2 − 3.46i)25-s + (−2.5 + 4.33i)29-s + 4.89·35-s − 5·37-s + (1 + 1.73i)41-s + (2.44 − 4.24i)43-s + (4.89 − 8.48i)47-s + (−8.49 − 14.7i)49-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.925 + 1.60i)7-s + (−0.738 + 1.27i)11-s + (−0.416 − 0.720i)13-s + 1.21·17-s − 1.12·19-s + (−0.510 − 0.884i)23-s + (0.400 − 0.692i)25-s + (−0.464 + 0.804i)29-s + 0.828·35-s − 0.821·37-s + (0.156 + 0.270i)41-s + (0.373 − 0.646i)43-s + (0.714 − 1.23i)47-s + (−1.21 − 2.10i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 2592,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4668436497\)
\(L(\frac12)\) \(\approx\) \(0.4668436497\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.44 - 4.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 + (2.44 + 4.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 + 4.24i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-4.89 - 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.44 - 4.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (-7.34 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.89 + 8.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13T + 89T^{2} \)
97 \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.663948278219009971746873540223, −8.057326933969779623724394170646, −7.14073078770360495107521177842, −6.30368969865971317400752194243, −5.43569521294470739955624883785, −4.96723940421515035031621918550, −3.77908121959890920146089578033, −2.69402587115598948954658671312, −2.10616853534755143257576921051, −0.17334210616363613287510515008, 1.02721855768322341959256142173, 2.62120343922177412206940566901, 3.62155663416698793260350889730, 3.96741155815515552995583893735, 5.25747796584549865370974928412, 6.13649653587669236916820187185, 6.85216811204924948019906319164, 7.58321376724861911629303585993, 8.085085932114656817830951653023, 9.251376817587219872978969365638

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