Properties

Label 2-2592-9.7-c1-0-9
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

\(L(1)\) \(\approx\) \(1.438584786\)
\(L(\frac12)\) \(\approx\) \(1.438584786\)
\(L(\frac{3}{2})\) 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

−9.240638411799817841076930018616, −8.769866488282790390209258241113, −7.56344535284593409193309137541, −6.76024950915706623828347744755, −6.21818028981716295465998060269, −5.63015407869931119851626258312, −4.47588204588043979425975694332, −3.46018423650292637169823981741, −2.39196590001365825149904160048, −2.01260364942884000205549923310, 0.50053411384022147267961610989, 1.19404009453426925572084244493, 2.89697892030605159100425723381, 3.56482838850761605662286957359, 4.64248549722738445903619632786, 5.37293241351200262661112254117, 6.21868148396660101326451739737, 6.88199092358257228179778997796, 7.904919919686346065718345805472, 8.619219736050969205478709358354

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