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Properties

Label 2-2592-72.13-c1-0-6
Degree $2$
Conductor $2592$
Sign $0.0871 - 0.996i$
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  + (−1.73 − i)5-s + (−1 − 1.73i)7-s + (−3.46 − 2i)13-s + 2·17-s + 4i·19-s + (−2 + 3.46i)23-s + (−0.500 − 0.866i)25-s + (−5.19 + 3i)29-s + (1 − 1.73i)31-s + 3.99i·35-s − 8i·37-s + (1 − 1.73i)41-s + (−3.46 + 2i)43-s + (6 + 10.3i)47-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (−0.774 − 0.447i)5-s + (−0.377 − 0.654i)7-s + (−0.960 − 0.554i)13-s + 0.485·17-s + 0.917i·19-s + (−0.417 + 0.722i)23-s + (−0.100 − 0.173i)25-s + (−0.964 + 0.557i)29-s + (0.179 − 0.311i)31-s + 0.676i·35-s − 1.31i·37-s + (0.156 − 0.270i)41-s + (−0.528 + 0.304i)43-s + (0.875 + 1.51i)47-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.0871 - 0.996i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.0871 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5774011546\)
\(L(\frac12)\) \(\approx\) \(0.5774011546\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.46 + 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.19 - 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.46 - 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.8 - 8i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.123311939848167710456037462000, −8.015267715421914169161718497897, −7.69368400492607586323723351292, −6.98172913785816902647132834809, −5.84798392586787939287042441282, −5.19210627212078447818302626559, −4.06236337361926667903528598385, −3.66586321795779185318488118654, −2.42415207043780896877837754664, −0.997657884103675985263414835213, 0.22324716217930014605368567332, 2.05851746693722343004634160854, 2.92907582058478549186454464523, 3.80911129102442660839329032250, 4.74310633057863681940705611054, 5.55552616377425691833469760638, 6.58811414850821443581073648366, 7.11280203875165215396627051854, 7.926893374071571835996178755869, 8.663139198667010309046767083846

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