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Properties

Label 2-2592-72.13-c1-0-0
Degree $2$
Conductor $2592$
Sign $-0.999 + 0.0123i$
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  + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)7-s + (−2.59 + 1.5i)11-s + (4.58 + 2.64i)13-s − 5.29·17-s − 5.29i·19-s + (−2.64 + 4.58i)23-s + (−2 − 3.46i)25-s + (5.19 − 3i)29-s + (−3.5 + 6.06i)31-s − 0.999i·35-s − 5.29i·37-s + (−2.64 + 4.58i)41-s + (−9.16 + 5.29i)43-s + (3 − 5.19i)49-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (0.188 + 0.327i)7-s + (−0.783 + 0.452i)11-s + (1.27 + 0.733i)13-s − 1.28·17-s − 1.21i·19-s + (−0.551 + 0.955i)23-s + (−0.400 − 0.692i)25-s + (0.964 − 0.557i)29-s + (−0.628 + 1.08i)31-s − 0.169i·35-s − 0.869i·37-s + (−0.413 + 0.715i)41-s + (−1.39 + 0.806i)43-s + (0.428 − 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.999 + 0.0123i$
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: \((2,\ 2592,\ (\ :1/2),\ -0.999 + 0.0123i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09657754691\)
\(L(\frac12)\) \(\approx\) \(0.09657754691\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.58 - 2.64i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + (2.64 - 4.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.19 + 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.29iT - 37T^{2} \)
41 \( 1 + (2.64 - 4.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.16 - 5.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + (3.46 + 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 3.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
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

−9.064768310878275141742170303272, −8.544740889078321305962215661124, −7.88309047387293185000115232021, −6.87339164117732663715866484812, −6.33716851995500793202092297073, −5.23046618634070937203282247815, −4.56039458289133017492077353642, −3.74078240587203922901239818808, −2.57877846619141349489150269374, −1.62060260981156723387388775869, 0.03091430045308278254473030860, 1.49257720046064312244207441231, 2.76220234475831489304915850563, 3.66778831681467163213491640747, 4.37739407692341192815059591326, 5.49487274942091809875671570692, 6.13878341316070335992822643076, 6.98892221220991029353975411524, 7.951043435695113757201322911154, 8.315765901382988526307521511667

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