Properties

Label 2-2592-9.7-c1-0-25
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  + (0.133 + 0.232i)5-s + (0.232 + 0.401i)13-s − 2.26·17-s + (2.46 − 4.26i)25-s + (3.33 − 5.76i)29-s + 11.3·37-s + (4 + 6.92i)41-s + (3.5 + 6.06i)49-s − 4·53-s + (2.69 − 4.66i)61-s + (−0.0621 + 0.107i)65-s + 10.8·73-s + (−0.303 − 0.526i)85-s − 16.6·89-s + (9 − 15.5i)97-s + ⋯
L(s)  = 1  + (0.0599 + 0.103i)5-s + (0.0643 + 0.111i)13-s − 0.550·17-s + (0.492 − 0.853i)25-s + (0.618 − 1.07i)29-s + 1.87·37-s + (0.624 + 1.08i)41-s + (0.5 + 0.866i)49-s − 0.549·53-s + (0.345 − 0.597i)61-s + (−0.00771 + 0.0133i)65-s + 1.27·73-s + (−0.0329 − 0.0570i)85-s − 1.76·89-s + (0.913 − 1.58i)97-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(1.784228456\)
\(L(\frac12)\) \(\approx\) \(1.784228456\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 - 0.232i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.232 - 0.401i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.33 + 5.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.69 + 4.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
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

−8.824196629112438657926659296095, −8.105037421503278577813367632949, −7.40285832230839749006444024521, −6.39189603245522908877559025181, −6.00186697089470922747680442243, −4.74159924378132057868790421317, −4.23470991645069879708163952933, −3.01073130148301822043233718155, −2.20394028021165459529155335462, −0.795281354231454063194466972172, 0.911550917391254516397338482955, 2.17574757971389090369059240755, 3.16856800590572243765022281199, 4.14207004787733053828312316234, 4.99525111679079482846731394702, 5.77688159139883416898236950329, 6.66356783425401116588300001509, 7.33700055694683035490818551218, 8.198382564483090137953311918375, 8.937652960494314496920686958902

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