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Properties

Label 2-2592-9.7-c1-0-23
Degree $2$
Conductor $2592$
Sign $0.939 - 0.342i$
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  + (0.5 + 0.866i)5-s + (−1.5 + 2.59i)7-s + (1.5 − 2.59i)11-s + 4·17-s + 6·19-s + (−3 − 5.19i)23-s + (2 − 3.46i)25-s + (1 − 1.73i)29-s + (−4.5 − 7.79i)31-s − 3·35-s − 2·37-s + (5 + 8.66i)41-s + (−3 + 5.19i)43-s + (−3 + 5.19i)47-s + (−1 − 1.73i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.452 − 0.783i)11-s + 0.970·17-s + 1.37·19-s + (−0.625 − 1.08i)23-s + (0.400 − 0.692i)25-s + (0.185 − 0.321i)29-s + (−0.808 − 1.39i)31-s − 0.507·35-s − 0.328·37-s + (0.780 + 1.35i)41-s + (−0.457 + 0.792i)43-s + (−0.437 + 0.757i)47-s + (−0.142 − 0.247i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
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: \((2,\ 2592,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.897930468\)
\(L(\frac12)\) \(\approx\) \(1.897930468\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13T + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
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

−9.009878536438654192747774344011, −8.178034473288664755472489123511, −7.46910633666343223508731390170, −6.31403745789581725271700967264, −6.03541624299120522457388249947, −5.19120973373606643036132078366, −4.00207924491918460708969905476, −3.06096059242366661000727043563, −2.42396559664415507346641130286, −0.906687015890173513004562255589, 0.889055183230549441424349587698, 1.85761310071153962994539198344, 3.46975209535711720735335886196, 3.73935937451302002525589975165, 5.14204958645387987872995266013, 5.47516009337414249314611186720, 6.86518062757684425288694262322, 7.14357525279656467066571354834, 7.958783444726759735940185394977, 9.037713468325235734551761831073

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