Properties

Label 2-2592-9.7-c1-0-31
Degree $2$
Conductor $2592$
Sign $0.984 + 0.173i$
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  + (1.32 + 2.29i)5-s + (2.18 − 3.79i)7-s + (1.79 − 3.10i)11-s + (3.29 + 5.70i)13-s − 1.73·17-s + 2.55·19-s + (−3.79 − 6.56i)23-s + (−1 + 1.73i)25-s + (3.05 − 5.29i)29-s + (4.37 + 7.58i)31-s + 11.5·35-s + 2.58·37-s + (−0.913 − 1.58i)41-s + (1.27 − 2.20i)43-s + (−4 + 6.92i)47-s + ⋯
L(s)  = 1  + (0.591 + 1.02i)5-s + (0.827 − 1.43i)7-s + (0.540 − 0.935i)11-s + (0.912 + 1.58i)13-s − 0.420·17-s + 0.585·19-s + (−0.790 − 1.36i)23-s + (−0.200 + 0.346i)25-s + (0.567 − 0.982i)29-s + (0.786 + 1.36i)31-s + 1.95·35-s + 0.424·37-s + (−0.142 − 0.247i)41-s + (0.194 − 0.336i)43-s + (−0.583 + 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488465999\)
\(L(\frac12)\) \(\approx\) \(2.488465999\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 2.29i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.18 + 3.79i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.79 + 3.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.29 - 5.70i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (3.79 + 6.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.05 + 5.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.37 - 7.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.58T + 37T^{2} \)
41 \( 1 + (0.913 + 1.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.708 + 1.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.27 - 2.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.417T + 71T^{2} \)
73 \( 1 + 6.16T + 73T^{2} \)
79 \( 1 + (-4.73 + 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.58 - 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-2.58 + 4.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.742659795071180465296034652844, −8.160077251119702403780627612253, −7.15911133133140005410406079210, −6.46886475964774863057890764304, −6.19048228044439656677777412790, −4.66983553288639943365279515967, −4.12295608144152529013089482859, −3.19511493515216364668654210775, −2.00275865081148178411037535077, −0.995671463335322164850564201097, 1.19878492507477652118408339795, 1.95199700473962754561141676851, 3.06182550820998718425793961640, 4.31406187966844855340646254434, 5.22263115984498959984805856983, 5.57927601469608645572964674564, 6.34179429251603972887766547439, 7.67964831196925221508555747294, 8.214610673068118005841316085692, 8.929420808541458156940505803846

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