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Properties

Label 2-2592-36.23-c1-0-45
Degree $2$
Conductor $2592$
Sign $-0.996 - 0.0871i$
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  + (−2.44 − 1.41i)5-s + (1.73 − i)7-s + (−1.41 − 2.44i)11-s + (1 − 1.73i)13-s − 6i·19-s + (−2.82 + 4.89i)23-s + (1.49 + 2.59i)25-s + (−2.44 + 1.41i)29-s + (−1.73 − i)31-s − 5.65·35-s + 6·37-s + (−4.89 − 2.82i)41-s + (1.73 − i)43-s + (5.65 + 9.79i)47-s + (−1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (−1.09 − 0.632i)5-s + (0.654 − 0.377i)7-s + (−0.426 − 0.738i)11-s + (0.277 − 0.480i)13-s − 1.37i·19-s + (−0.589 + 1.02i)23-s + (0.299 + 0.519i)25-s + (−0.454 + 0.262i)29-s + (−0.311 − 0.179i)31-s − 0.956·35-s + 0.986·37-s + (−0.765 − 0.441i)41-s + (0.264 − 0.152i)43-s + (0.825 + 1.42i)47-s + (−0.214 + 0.371i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.996 - 0.0871i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.996 - 0.0871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5567741833\)
\(L(\frac12)\) \(\approx\) \(0.5567741833\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.41 + 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 + i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (4.89 + 2.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 + (1.41 - 2.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (12.1 - 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.41 - 2.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.456652958308280906174312411893, −7.69154956947244016663041954812, −7.33735783989996509081506526871, −6.05756934379817628431826042092, −5.25288838418501413225877521319, −4.47017580663652051024416555828, −3.75453584965584673818776003744, −2.77255645348315540028526384215, −1.27500338240728386290810802776, −0.19407022268631289996401194037, 1.67343580243633774692765711856, 2.65811166387654362206194989230, 3.81186167162098117519020802128, 4.34145713729447033859388188520, 5.35611009282792973236087598672, 6.26829322203942052996599000949, 7.13031519871246856374350018087, 7.83894552972783360056167115418, 8.261766991701105583078462041689, 9.179782866817352342484414291531

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