Properties

Label 1-18e2-324.239-r0-0-0
Degree $1$
Conductor $324$
Sign $-0.516 + 0.856i$
Analytic cond. $1.50464$
Root an. cond. $1.50464$
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  + (−0.597 + 0.802i)5-s + (0.835 + 0.549i)7-s + (−0.993 + 0.116i)11-s + (−0.286 + 0.957i)13-s + (−0.173 − 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (0.686 − 0.727i)29-s + (0.0581 + 0.998i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)37-s + (−0.973 + 0.230i)41-s + (−0.396 + 0.918i)43-s + (−0.0581 + 0.998i)47-s + ⋯
L(s)  = 1  + (−0.597 + 0.802i)5-s + (0.835 + 0.549i)7-s + (−0.993 + 0.116i)11-s + (−0.286 + 0.957i)13-s + (−0.173 − 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (0.686 − 0.727i)29-s + (0.0581 + 0.998i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)37-s + (−0.973 + 0.230i)41-s + (−0.396 + 0.918i)43-s + (−0.0581 + 0.998i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.516 + 0.856i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ -0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4141124314 + 0.7336000313i\)
\(L(\frac12)\) \(\approx\) \(0.4141124314 + 0.7336000313i\)
\(L(1)\) \(\approx\) \(0.8105909293 + 0.3237303297i\)
\(L(1)\) \(\approx\) \(0.8105909293 + 0.3237303297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.597 + 0.802i)T \)
7 \( 1 + (0.835 + 0.549i)T \)
11 \( 1 + (-0.993 + 0.116i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.835 + 0.549i)T \)
29 \( 1 + (0.686 - 0.727i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.973 + 0.230i)T \)
43 \( 1 + (-0.396 + 0.918i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.993 - 0.116i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (0.686 + 0.727i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.973 - 0.230i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.597 + 0.802i)T \)
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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

−24.46426167271711029963160038528, −23.96910917699848414356443321607, −23.31385094510962380325947906267, −22.08711050078084162005089964525, −21.08267439059406213534532308830, −20.28028046265202226521053408848, −19.71617518626804511527659126704, −18.44826851403418798026284687392, −17.500852436685439750621322208454, −16.78118733434862978191672122574, −15.612820946128673330423922167383, −15.04456356963738602352401407227, −13.69271250130994020219009143893, −12.89947426056885203212152854682, −11.97822891081431594482256448631, −10.86909077164537690482637426417, −10.143074960886352103712160920587, −8.488016119080931955356415338039, −8.145374787303279116237980663119, −7.0260199672698057077572480841, −5.42371918605087739738555412134, −4.71436362617038456931509290533, −3.59453078046682864306585435858, −2.05304394735298389408973167212, −0.51929317648543470095700516434, 1.90165414572303397667711427930, 2.93053436895733720471955930683, 4.280720881937608252906119139548, 5.29001104598650491301044381118, 6.57784821633313525202921976772, 7.637816666008697733252902513111, 8.346806780292050846857384470996, 9.70306862938635263107472653701, 10.71274028170020158772951730191, 11.66042011760215840761023940169, 12.25530645976657341191797911397, 13.84280002073711378108812867533, 14.4349807952602093433472816692, 15.50449315072888890380202082450, 16.09815961067290655437500841313, 17.52220927633083339449721074953, 18.37487885078617506833722923621, 18.90365784481427914134598473911, 20.02850128510025232390418427939, 21.11436443850098044607688602847, 21.72312934767571657683293547994, 22.87952185720745398753587287369, 23.54434242767971954683678243030, 24.43967319714337410960651594152, 25.40820019864271970284629372973

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