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Properties

Label 2-2592-9.4-c1-0-14
Degree $2$
Conductor $2592$
Sign $-0.173 - 0.984i$
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 + 1.73i)5-s + (1.5 + 2.59i)7-s + (3 + 5.19i)11-s + (1.5 − 2.59i)13-s − 2·17-s + 3·19-s + (3 − 5.19i)23-s + (0.500 + 0.866i)25-s + (4 + 6.92i)29-s − 6·35-s + 7·37-s + (−4 + 6.92i)41-s + (−6 − 10.3i)43-s + (3 + 5.19i)47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.566 + 0.981i)7-s + (0.904 + 1.56i)11-s + (0.416 − 0.720i)13-s − 0.485·17-s + 0.688·19-s + (0.625 − 1.08i)23-s + (0.100 + 0.173i)25-s + (0.742 + 1.28i)29-s − 1.01·35-s + 1.15·37-s + (−0.624 + 1.08i)41-s + (−0.914 − 1.58i)43-s + (0.437 + 0.757i)47-s + (−0.142 + 0.247i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892013161\)
\(L(\frac12)\) \(\approx\) \(1.892013161\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (4 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.891667103155716382087337568851, −8.500865303779837356527538259968, −7.36844600912264891053400855178, −6.96634118827827064162849464350, −6.09741577203547591242424465180, −5.07066646026319733799880558189, −4.43729049925276825025392590420, −3.30102469354888504938231680871, −2.49525301079519857485053380653, −1.38565204603463967075572696870, 0.72387144762728601049807936173, 1.41979953575957238649747144745, 3.08153343481860954322300187124, 4.05159505793556041382238428323, 4.45837459255601040058079262012, 5.55730025082324528431218020971, 6.37336752372904864037437477029, 7.21219828811602097362528959425, 8.028887110238818119419288197843, 8.655604439665148787117898009333

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