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Properties

Label 1-72e2-5184.491-r0-0-0
Degree $1$
Conductor $5184$
Sign $0.172 + 0.985i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
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  + (−0.775 − 0.631i)5-s + (−0.929 + 0.369i)7-s + (0.810 + 0.585i)11-s + (−0.827 − 0.561i)13-s + (0.342 − 0.939i)17-s + (−0.675 − 0.737i)19-s + (−0.929 − 0.369i)23-s + (0.202 + 0.979i)25-s + (−0.328 − 0.944i)29-s + (−0.993 − 0.116i)31-s + (0.953 + 0.300i)35-s + (0.300 + 0.953i)37-s + (0.949 − 0.314i)41-s + (−0.912 + 0.409i)43-s + (0.116 + 0.993i)47-s + ⋯
L(s)  = 1  + (−0.775 − 0.631i)5-s + (−0.929 + 0.369i)7-s + (0.810 + 0.585i)11-s + (−0.827 − 0.561i)13-s + (0.342 − 0.939i)17-s + (−0.675 − 0.737i)19-s + (−0.929 − 0.369i)23-s + (0.202 + 0.979i)25-s + (−0.328 − 0.944i)29-s + (−0.993 − 0.116i)31-s + (0.953 + 0.300i)35-s + (0.300 + 0.953i)37-s + (0.949 − 0.314i)41-s + (−0.912 + 0.409i)43-s + (0.116 + 0.993i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.172 + 0.985i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ 0.172 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2609305315 + 0.2192163145i\)
\(L(\frac12)\) \(\approx\) \(0.2609305315 + 0.2192163145i\)
\(L(1)\) \(\approx\) \(0.6793925143 - 0.09133261472i\)
\(L(1)\) \(\approx\) \(0.6793925143 - 0.09133261472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.775 - 0.631i)T \)
7 \( 1 + (-0.929 + 0.369i)T \)
11 \( 1 + (0.810 + 0.585i)T \)
13 \( 1 + (-0.827 - 0.561i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.675 - 0.737i)T \)
23 \( 1 + (-0.929 - 0.369i)T \)
29 \( 1 + (-0.328 - 0.944i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
37 \( 1 + (0.300 + 0.953i)T \)
41 \( 1 + (0.949 - 0.314i)T \)
43 \( 1 + (-0.912 + 0.409i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (0.159 - 0.987i)T \)
61 \( 1 + (-0.858 - 0.512i)T \)
67 \( 1 + (-0.435 - 0.900i)T \)
71 \( 1 + (-0.573 + 0.819i)T \)
73 \( 1 + (-0.573 - 0.819i)T \)
79 \( 1 + (-0.448 - 0.893i)T \)
83 \( 1 + (-0.756 - 0.653i)T \)
89 \( 1 + (0.819 - 0.573i)T \)
97 \( 1 + (0.286 + 0.957i)T \)
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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

−17.92975200180726064481781434140, −16.89417939330750632179510637775, −16.53888729120223274828039799420, −16.03323220348095094957464957695, −14.99541408859453068518859421874, −14.58560882666800558125507697036, −14.05118942458627956708566040359, −13.07816668715476720352059989633, −12.41508709367805855820889075781, −11.88160802289853571664076267894, −11.12115504274811380506712683965, −10.40559423824574454327802964300, −9.89204670953847319748709985085, −8.99703530682675052856828869403, −8.36225162815282223164006934736, −7.39786993166082076076315824265, −7.02898811016790884197218877188, −6.188871626846975265992865865754, −5.678815024312805898803035566710, −4.33049475653430265499290917396, −3.79766058896441114426673942375, −3.38901561835228909633406896044, −2.345365571386032548201624178197, −1.44670385209003375682185714970, −0.13298616973649005847913848979, 0.657468684891314291015443079688, 1.864226207198390147855437036437, 2.748355303757882468654674584934, 3.45817307820554955889732428716, 4.38521532498938473704796981225, 4.79250283388662901707211592242, 5.8185759042201226139557547577, 6.472476094482602404845901983430, 7.38195616172589250376285485413, 7.775570071304930095261445130701, 8.80512138594087052869090966692, 9.36056215317093449769956383922, 9.83855939176434384757792355774, 10.75722611223584199043936677392, 11.84701501952053582510735020593, 11.940703568102689314708178122060, 12.8346220333466437372816278549, 13.1747119114193190729219555598, 14.29587299521520299774920846654, 14.902447482590179485636926505, 15.59298551825351286524694588863, 16.0417117743837809825212923987, 16.88153035828272682967010637957, 17.22623994029055828609879589886, 18.20610546402327246953911278525

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