Properties

Label 2-2592-36.23-c1-0-13
Degree $2$
Conductor $2592$
Sign $0.573 - 0.819i$
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  + (−0.896 − 0.517i)5-s + (0.232 − 0.133i)7-s + (1.93 + 3.34i)11-s + (1.23 − 2.13i)13-s + 6.69i·17-s + 1.73i·19-s + (2.96 − 5.13i)23-s + (−1.96 − 3.40i)25-s + (1.79 − 1.03i)29-s + (0.464 + 0.267i)31-s − 0.277·35-s − 6.46·37-s + (−1.79 − 1.03i)41-s + (−6.46 + 3.73i)43-s + (4.76 + 8.24i)47-s + ⋯
L(s)  = 1  + (−0.400 − 0.231i)5-s + (0.0877 − 0.0506i)7-s + (0.582 + 1.00i)11-s + (0.341 − 0.591i)13-s + 1.62i·17-s + 0.397i·19-s + (0.618 − 1.07i)23-s + (−0.392 − 0.680i)25-s + (0.332 − 0.192i)29-s + (0.0833 + 0.0481i)31-s − 0.0468·35-s − 1.06·37-s + (−0.280 − 0.161i)41-s + (−0.985 + 0.569i)43-s + (0.694 + 1.20i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.573 - 0.819i$
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: \((2,\ 2592,\ (\ :1/2),\ 0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559161242\)
\(L(\frac12)\) \(\approx\) \(1.559161242\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.896 + 0.517i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.232 + 0.133i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.93 - 3.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.23 + 2.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-2.96 + 5.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.79 + 1.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 + (1.79 + 1.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.46 - 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.76 - 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + (-3.72 + 6.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 8.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.42 + 4.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 + (-13.1 + 7.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.86 - 6.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.863468902751528402251177370792, −8.221535210429632203414386261626, −7.63364982512500407001640624444, −6.58442365327073346901176609950, −6.08318271281943835019648069292, −4.91505557811243271366915015944, −4.23693471496183546310375149968, −3.46421133629321887354383427454, −2.19706150752174516827524412933, −1.11869130187442382643334747618, 0.59613998304989793285109488810, 1.92653141635973633019535444798, 3.26999931223329682549610642879, 3.68212780100359735086073655281, 4.97343711081643292951300631632, 5.50584258806423766865876048855, 6.82832486917693826985799545425, 6.95065120251106212792685590803, 8.080215646602934115237551759628, 8.806987114846982341577760005582

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