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Properties

Label 2-2592-36.23-c1-0-26
Degree $2$
Conductor $2592$
Sign $0.906 - 0.422i$
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  + (2.72 + 1.57i)5-s + (2.98 − 1.72i)7-s + (2.28 + 3.94i)11-s + (3.44 − 5.97i)13-s + 3.46i·17-s + 4.89i·19-s + (−1.41 + 2.44i)23-s + (2.44 + 4.24i)25-s + (1.89 − 1.09i)29-s + (2.20 + 1.27i)31-s + 10.8·35-s − 4.89·37-s + (−3.55 − 2.04i)41-s + (2.51 − 1.44i)43-s + (1.09 + 1.89i)47-s + ⋯
L(s)  = 1  + (1.21 + 0.703i)5-s + (1.12 − 0.651i)7-s + (0.687 + 1.19i)11-s + (0.956 − 1.65i)13-s + 0.840i·17-s + 1.12i·19-s + (−0.294 + 0.510i)23-s + (0.489 + 0.848i)25-s + (0.352 − 0.203i)29-s + (0.396 + 0.229i)31-s + 1.83·35-s − 0.805·37-s + (−0.554 − 0.320i)41-s + (0.382 − 0.221i)43-s + (0.159 + 0.276i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.906 - 0.422i$
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: \((2,\ 2592,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.928771647\)
\(L(\frac12)\) \(\approx\) \(2.928771647\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.72 - 1.57i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.98 + 1.72i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.44 + 5.97i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + (1.41 - 2.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.89 + 1.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.20 - 1.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + (3.55 + 2.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.51 + 1.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 - 1.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 + (1.09 - 1.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.9 + 7.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 7.89T + 73T^{2} \)
79 \( 1 + (-1.73 + i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.20 - 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.02iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.900598478262001245661354086839, −8.059042516650480313922722379094, −7.50585103327974714535828847552, −6.51453346621777932987939590084, −5.90359598855412113947051902584, −5.13962363823406495583375090109, −4.10289765003949267391626478685, −3.25399911498659486258832417764, −1.90366910308007887423415309903, −1.39460184917397311088589556024, 1.14252676869518977892311054143, 1.84543232434134857179203904316, 2.89199085155089760384074389969, 4.32278652151294484969864122854, 4.87049351204301223439478291772, 5.83400775167996110506647836365, 6.26837307215168994694009663425, 7.22461301991440738629055697731, 8.497844115551179746994536415886, 8.999777488374907089815079758368

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