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Properties

Label 1-12e3-1728.715-r1-0-0
Degree $1$
Conductor $1728$
Sign $0.662 - 0.749i$
Analytic cond. $185.699$
Root an. cond. $185.699$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.976 − 0.216i)5-s + (−0.0871 − 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (0.737 − 0.675i)29-s + (0.766 − 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (0.906 − 0.422i)41-s + (−0.537 + 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯
L(s)  = 1  + (−0.976 − 0.216i)5-s + (−0.0871 − 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (0.737 − 0.675i)29-s + (0.766 − 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (0.906 − 0.422i)41-s + (−0.537 + 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ 0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.545414989 - 0.6968347582i\)
\(L(\frac12)\) \(\approx\) \(1.545414989 - 0.6968347582i\)
\(L(1)\) \(\approx\) \(0.9363875641 - 0.1672528677i\)
\(L(1)\) \(\approx\) \(0.9363875641 - 0.1672528677i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.976 - 0.216i)T \)
7 \( 1 + (-0.0871 - 0.996i)T \)
11 \( 1 + (0.537 - 0.843i)T \)
13 \( 1 + (0.0436 + 0.999i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.991 + 0.130i)T \)
23 \( 1 + (0.0871 - 0.996i)T \)
29 \( 1 + (0.737 - 0.675i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.991 + 0.130i)T \)
41 \( 1 + (0.906 - 0.422i)T \)
43 \( 1 + (-0.537 + 0.843i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.976 + 0.216i)T \)
61 \( 1 + (0.953 + 0.300i)T \)
67 \( 1 + (0.999 - 0.0436i)T \)
71 \( 1 + (-0.258 + 0.965i)T \)
73 \( 1 + (0.258 + 0.965i)T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (0.675 + 0.737i)T \)
89 \( 1 + (0.965 - 0.258i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.03375374905338669479691600045, −19.58544733564663888184202579116, −18.78215559782545830616879874791, −17.767572472180900331018886770161, −17.6964365250507380662011996806, −16.16705306817319138400890704398, −15.70059510461458380351249574876, −15.18650574020497765201306831638, −14.43816375934241774260973326417, −13.44561901770215099944480854013, −12.49495072791142529254316901118, −11.96461108584196735221900651939, −11.37573428793350179485753975043, −10.41681914999669940564358308362, −9.51115785832504356816489503662, −8.769158584340839661744083541581, −7.986850497987075858652881577623, −7.148390254904454614424633116423, −6.490746027984934262533048586186, −5.26450196062093855679189855867, −4.76322501428931789273996181043, −3.52198114937987259733477246807, −2.96707270165308394808119886595, −1.86173862602397428098707096567, −0.6264554808377148522951743270, 0.53836040172595738732445359043, 1.2453071019085411028878874782, 2.64530919455029316205669979078, 3.77026778181902444929400472760, 4.15248004321545579980615317439, 5.01783067922776828132943815809, 6.4429549586451715380359155969, 6.7818299907806113478596124697, 7.885359016030006704324312527089, 8.43567938306408419707797766904, 9.31911694378500766659079289573, 10.23230486589215356275911198891, 11.24931630942608522746387288494, 11.491435696640056747818181132041, 12.48265147558940556706192404204, 13.3563865879875855393379505827, 14.077549758204925569378386010624, 14.67399672452718000727117133259, 15.897809578348545918903470694422, 16.14826559398359313468375965716, 16.993308497275367522855706340609, 17.63255307187789720989331542968, 18.859867812414009582672247350272, 19.25855898612132915408532662403, 19.930222495254509164800277873204

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