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Properties

Label 2-2592-36.11-c1-0-12
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  + (−1.22 + 0.707i)5-s + (−3.46 − 2i)7-s + (−2.82 + 4.89i)11-s + (−2 − 3.46i)13-s − 4.24i·17-s + (2.82 + 4.89i)23-s + (−1.50 + 2.59i)25-s + (−1.22 − 0.707i)29-s + (3.46 − 2i)31-s + 5.65·35-s − 6·37-s + (8.57 − 4.94i)41-s + (6.92 + 4i)43-s + (2.82 − 4.89i)47-s + (4.49 + 7.79i)49-s + ⋯
L(s)  = 1  + (−0.547 + 0.316i)5-s + (−1.30 − 0.755i)7-s + (−0.852 + 1.47i)11-s + (−0.554 − 0.960i)13-s − 1.02i·17-s + (0.589 + 1.02i)23-s + (−0.300 + 0.519i)25-s + (−0.227 − 0.131i)29-s + (0.622 − 0.359i)31-s + 0.956·35-s − 0.986·37-s + (1.33 − 0.773i)41-s + (1.05 + 0.609i)43-s + (0.412 − 0.714i)47-s + (0.642 + 1.11i)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: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.996 - 0.0871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9089664611\)
\(L(\frac12)\) \(\approx\) \(0.9089664611\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.82 - 4.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.82 - 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 + 0.707i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.46 + 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-8.57 + 4.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.92 - 4i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.82 + 4.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.82 + 4.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.098499128823278060174845813988, −7.70413542665282968771809498922, −7.38957759733281850635862519024, −6.94136876377116466900698716607, −5.73810332577250698005293603046, −4.96074944553709200276529476419, −3.99518700556117790613915316426, −3.16942193121370375865020008138, −2.38925882220809031849334686878, −0.58430327703161209269963136912, 0.55496544197686139230712650916, 2.34434482235121156757902127143, 3.10805822210596765374115347187, 3.98786445992822789473056402991, 4.95202032383105110986669775212, 6.00391088818686120730755592987, 6.33043114021776538443489196145, 7.38066093503607038487084289927, 8.314439922513576408238453633746, 8.786193988308417371193618735471

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