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Properties

Label 2-2592-9.4-c1-0-33
Degree $2$
Conductor $2592$
Sign $-0.766 + 0.642i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)5-s + (−3 + 5.19i)13-s − 2·17-s + (0.500 + 0.866i)25-s + (−5 − 8.66i)29-s − 2·37-s + (5 − 8.66i)41-s + (3.5 − 6.06i)49-s − 14·53-s + (5 + 8.66i)61-s + (−6 − 10.3i)65-s − 6·73-s + (2 − 3.46i)85-s − 10·89-s + (−9 − 15.5i)97-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (−0.832 + 1.44i)13-s − 0.485·17-s + (0.100 + 0.173i)25-s + (−0.928 − 1.60i)29-s − 0.328·37-s + (0.780 − 1.35i)41-s + (0.5 − 0.866i)49-s − 1.92·53-s + (0.640 + 1.10i)61-s + (−0.744 − 1.28i)65-s − 0.702·73-s + (0.216 − 0.375i)85-s − 1.05·89-s + (−0.913 − 1.58i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
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: \(1\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (9 + 15.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.643592875549318202947190257217, −7.59770541603171538995517470939, −7.13284103448834137323053168164, −6.44597021406819334855555102172, −5.50467918575583979232735301346, −4.42553277927238074097862508984, −3.85023463839598664574010788232, −2.70080902117197413099128563541, −1.87434943159650414500715140578, 0, 1.24908843676189811929719496116, 2.62637205874520343167341067342, 3.51015662420482243419935972866, 4.61522750701452278382929100407, 5.12197879180080150589992337435, 6.00013225684640725720855123249, 7.02116844692040028354355110798, 7.78945872239334069432400870310, 8.322147529835774295258516687193

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