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Properties

Label 1-18e2-324.7-r1-0-0
Degree $1$
Conductor $324$
Sign $-0.875 - 0.483i$
Analytic cond. $34.8186$
Root an. cond. $34.8186$
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  + (0.396 − 0.918i)5-s + (0.0581 + 0.998i)7-s + (−0.597 + 0.802i)11-s + (−0.686 + 0.727i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.0581 − 0.998i)23-s + (−0.686 − 0.727i)25-s + (0.973 − 0.230i)29-s + (−0.893 + 0.448i)31-s + (0.939 + 0.342i)35-s + (−0.939 + 0.342i)37-s + (−0.286 − 0.957i)41-s + (0.993 + 0.116i)43-s + (−0.893 − 0.448i)47-s + ⋯
L(s)  = 1  + (0.396 − 0.918i)5-s + (0.0581 + 0.998i)7-s + (−0.597 + 0.802i)11-s + (−0.686 + 0.727i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.0581 − 0.998i)23-s + (−0.686 − 0.727i)25-s + (0.973 − 0.230i)29-s + (−0.893 + 0.448i)31-s + (0.939 + 0.342i)35-s + (−0.939 + 0.342i)37-s + (−0.286 − 0.957i)41-s + (0.993 + 0.116i)43-s + (−0.893 − 0.448i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.875 - 0.483i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ -0.875 - 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1398376201 - 0.5428829631i\)
\(L(\frac12)\) \(\approx\) \(0.1398376201 - 0.5428829631i\)
\(L(1)\) \(\approx\) \(0.8806904071 - 0.1089080714i\)
\(L(1)\) \(\approx\) \(0.8806904071 - 0.1089080714i\)

Euler product

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

−25.43990101335457589406972966956, −24.26301247110467890104112701575, −23.409573345339491821110745213157, −22.64625047402991092883647371913, −21.65075717021455203736247353717, −20.965833149168630819195074228748, −19.74146265240673182088251354895, −19.06689973828600345372227524505, −17.967982856448670334306679375580, −17.269479011324267839366716431778, −16.30326403292543098053203442765, −15.11776244649604253327234522784, −14.304370490889561477706107391553, −13.50840187094753348328464951961, −12.5467189185934964773174979747, −11.107693835719862794435373045139, −10.48429513397163678406473629826, −9.75771943975596044072992054376, −8.14253101411256064929392194700, −7.43643060691310135611246970786, −6.280351034994501501827718560274, −5.35239266201973099580256248478, −3.81203679133687378702347406778, −2.95557912060916620226258507468, −1.501972788747124934383087908502, 0.15317824490962791892028393665, 1.85606734290501837657879216346, 2.72944836384598484430322299670, 4.704123745065339446644038600862, 5.0574780659478560780450802389, 6.40879074733482813417169257163, 7.569476710374779323289300230624, 8.86015853945160718628983868141, 9.33488996454923342821713306439, 10.48886512428341051163367053262, 11.97188704525168898212376710741, 12.37921468661939625615430993133, 13.455916201588943533073200890466, 14.49084079816570302821514266469, 15.563506904726778178377095286325, 16.28507842111578705628336092277, 17.38354005058033143685329159439, 18.11353184429882826985801668051, 19.12069738073284583079169792365, 20.17064428616378129348643261987, 20.9853074935401816795410724418, 21.71317807311602119427377376816, 22.65246598200611920042576091979, 23.831291325421155183502038411874, 24.52514010091412962372131381308

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