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Properties

Label 1-72e2-5184.1067-r0-0-0
Degree $1$
Conductor $5184$
Sign $-0.501 + 0.865i$
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  + (0.944 + 0.328i)5-s + (−0.524 − 0.851i)7-s + (−0.561 − 0.827i)11-s + (−0.272 − 0.962i)13-s + (0.642 + 0.766i)17-s + (−0.300 + 0.953i)19-s + (−0.524 + 0.851i)23-s + (0.784 + 0.620i)25-s + (−0.858 − 0.512i)29-s + (−0.286 + 0.957i)31-s + (−0.216 − 0.976i)35-s + (−0.976 − 0.216i)37-s + (0.929 + 0.369i)41-s + (0.997 − 0.0726i)43-s + (−0.957 + 0.286i)47-s + ⋯
L(s)  = 1  + (0.944 + 0.328i)5-s + (−0.524 − 0.851i)7-s + (−0.561 − 0.827i)11-s + (−0.272 − 0.962i)13-s + (0.642 + 0.766i)17-s + (−0.300 + 0.953i)19-s + (−0.524 + 0.851i)23-s + (0.784 + 0.620i)25-s + (−0.858 − 0.512i)29-s + (−0.286 + 0.957i)31-s + (−0.216 − 0.976i)35-s + (−0.976 − 0.216i)37-s + (0.929 + 0.369i)41-s + (0.997 − 0.0726i)43-s + (−0.957 + 0.286i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ -0.501 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3326921071 + 0.5770469487i\)
\(L(\frac12)\) \(\approx\) \(0.3326921071 + 0.5770469487i\)
\(L(1)\) \(\approx\) \(0.9588217976 + 0.004240827159i\)
\(L(1)\) \(\approx\) \(0.9588217976 + 0.004240827159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.944 + 0.328i)T \)
7 \( 1 + (-0.524 - 0.851i)T \)
11 \( 1 + (-0.561 - 0.827i)T \)
13 \( 1 + (-0.272 - 0.962i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (-0.300 + 0.953i)T \)
23 \( 1 + (-0.524 + 0.851i)T \)
29 \( 1 + (-0.858 - 0.512i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
37 \( 1 + (-0.976 - 0.216i)T \)
41 \( 1 + (0.929 + 0.369i)T \)
43 \( 1 + (0.997 - 0.0726i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (0.187 + 0.982i)T \)
61 \( 1 + (0.355 - 0.934i)T \)
67 \( 1 + (0.244 - 0.969i)T \)
71 \( 1 + (-0.422 - 0.906i)T \)
73 \( 1 + (-0.422 + 0.906i)T \)
79 \( 1 + (-0.918 - 0.396i)T \)
83 \( 1 + (-0.999 - 0.0145i)T \)
89 \( 1 + (-0.906 - 0.422i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−17.758320212536293113895562869950, −17.18596777131504754727429919371, −16.23714085805132415271357484452, −16.06612493426176190411014903520, −14.98699759454496373548722686232, −14.49961210900951168063365984557, −13.75600093977061495453135406571, −12.905827580892854820610965080459, −12.6454809551724300558688703230, −11.82449210759365360552033539007, −11.11047683425849219623518674402, −10.03535280098527265195779208901, −9.72710308205625414411728633572, −9.04412622260085138566796944066, −8.50418208183062476501020894676, −7.34961810445016215686274224395, −6.82654093504302282898606677764, −6.01133795931704042225084288895, −5.32902316672009048502767018984, −4.78468038564895234342199996905, −3.88939076922945924173622437291, −2.59266487369099671099614327908, −2.400239426461873857587128380313, −1.52354599408302552601283181001, −0.16515628445850851267812977054, 1.10302668033534822261529070517, 1.838320253492528598583421629400, 2.94056457251135214037999825747, 3.39108368375825887336103864445, 4.19891272243554931472511317937, 5.46484585349128661495721239946, 5.74571863189504316509872995378, 6.41436963376174556038539007839, 7.44268950827094400244816077203, 7.8384962297118947131923335735, 8.75771768679173638104470968990, 9.65507552629753553073798407389, 10.2295497851379162313734596951, 10.59262739912178534073715416164, 11.297423688022533530920852090805, 12.581333353608841688526593098919, 12.7893932277691347029550226896, 13.645689206410088200143132863035, 14.11368053967096788142451096312, 14.74461565296906685045815640598, 15.63738540913136058728718716247, 16.29865770165753913028439600279, 16.98423406581783261998286060568, 17.45464907662023010715314186663, 18.145493633008897387820018094235

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