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Properties

Label 2-2592-36.11-c1-0-37
Degree $2$
Conductor $2592$
Sign $-0.422 + 0.906i$
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  + (0.275 − 0.158i)5-s + (1.25 + 0.724i)7-s + (0.548 − 0.949i)11-s + (−1.44 − 2.51i)13-s − 3.46i·17-s − 4.89i·19-s + (−1.41 − 2.44i)23-s + (−2.44 + 4.24i)25-s + (−7.89 − 4.56i)29-s + (−6.45 + 3.72i)31-s + 0.460·35-s + 4.89·37-s + (−8.44 + 4.87i)41-s + (5.97 + 3.44i)43-s + (4.56 − 7.89i)47-s + ⋯
L(s)  = 1  + (0.123 − 0.0710i)5-s + (0.474 + 0.273i)7-s + (0.165 − 0.286i)11-s + (−0.402 − 0.696i)13-s − 0.840i·17-s − 1.12i·19-s + (−0.294 − 0.510i)23-s + (−0.489 + 0.848i)25-s + (−1.46 − 0.846i)29-s + (−1.15 + 0.668i)31-s + 0.0778·35-s + 0.805·37-s + (−1.31 + 0.761i)41-s + (0.911 + 0.526i)43-s + (0.665 − 1.15i)47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.173845126\)
\(L(\frac12)\) \(\approx\) \(1.173845126\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.275 + 0.158i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.25 - 0.724i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.548 + 0.949i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 + 2.51i)T + (-6.5 + 11.2i)T^{2} \)
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 \( 1 + (7.89 + 4.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.45 - 3.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + (8.44 - 4.87i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.97 - 3.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.56 + 7.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.41iT - 53T^{2} \)
59 \( 1 + (4.56 + 7.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.41 + 2.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.56T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + (1.73 + i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.93 + 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
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

−8.728988212251114871552660097972, −7.78163112790830955845844335225, −7.27476730502115130177150719058, −6.27992545094666573499287915535, −5.38965078231367033370850685857, −4.88334116866081475064170604968, −3.76093448880219826047195965114, −2.79054597002352348861306772571, −1.82572965913131229714887071100, −0.36743271443800158687692803306, 1.51618139158738222713680947617, 2.22853168912669476104606924208, 3.74811701744000138315218234987, 4.15640849969852949799537078072, 5.32709159530813921412319960451, 5.97144608792493486915759353831, 6.90415748468304377068972648921, 7.66848334729301630469015280563, 8.215797682799329984802276208189, 9.300225922126490517920411403252

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