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Properties

Label 1-1323-1323.394-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.0901 + 0.995i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.411 + 0.911i)2-s + (−0.661 − 0.749i)4-s + (0.921 + 0.388i)5-s + (0.955 − 0.294i)8-s + (−0.733 + 0.680i)10-s + (−0.583 − 0.811i)11-s + (0.270 + 0.962i)13-s + (−0.124 + 0.992i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.318 − 0.947i)20-s + (0.980 − 0.198i)22-s + (0.980 − 0.198i)23-s + (0.698 + 0.715i)25-s + (−0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)2-s + (−0.661 − 0.749i)4-s + (0.921 + 0.388i)5-s + (0.955 − 0.294i)8-s + (−0.733 + 0.680i)10-s + (−0.583 − 0.811i)11-s + (0.270 + 0.962i)13-s + (−0.124 + 0.992i)16-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (−0.318 − 0.947i)20-s + (0.980 − 0.198i)22-s + (0.980 − 0.198i)23-s + (0.698 + 0.715i)25-s + (−0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0901 + 0.995i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.0901 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003230872 + 0.9165563330i\)
\(L(\frac12)\) \(\approx\) \(1.003230872 + 0.9165563330i\)
\(L(1)\) \(\approx\) \(0.8796600414 + 0.4417375262i\)
\(L(1)\) \(\approx\) \(0.8796600414 + 0.4417375262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.411 + 0.911i)T \)
5 \( 1 + (0.921 + 0.388i)T \)
11 \( 1 + (-0.583 - 0.811i)T \)
13 \( 1 + (0.270 + 0.962i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.980 - 0.198i)T \)
29 \( 1 + (-0.661 + 0.749i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.921 + 0.388i)T \)
43 \( 1 + (-0.797 - 0.603i)T \)
47 \( 1 + (-0.583 - 0.811i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (0.542 + 0.840i)T \)
61 \( 1 + (-0.661 + 0.749i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.365 + 0.930i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.270 - 0.962i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−20.99589280433603523446751203212, −19.98018818775436367185901491264, −19.41204310034697831779485406055, −18.39928443385433676641830517950, −17.66517121954513286996256557817, −17.37208876951847532679723651735, −16.46569044217214726815272628033, −15.3995225224923246482450205184, −14.531201897958583847350284631680, −13.473322072405166984423682401736, −12.755290490505906639740759964349, −12.64639069012463525563303241219, −11.21423218241701550993804203170, −10.609957506746928377008657773382, −9.87118386754748280867206075716, −9.223303078885138697563825078260, −8.34212327723608165210442700867, −7.60933760655953993144231490035, −6.43016477964753503027821636254, −5.305455622926488062447585468497, −4.703328040111980032179805218752, −3.525106733240472370080521950843, −2.5536676651603530534610756698, −1.81515690801992258745086398649, −0.7618347484475014602673999196, 0.9967200602258982931687860453, 2.07813804901963758183628448264, 3.22573954329327789022814447492, 4.47615785732239323259708138179, 5.43530921406515674648215428477, 6.03593287579050558880712236869, 6.831759713792631136007010827213, 7.60932799597140615246181131896, 8.64267987048556834607325335715, 9.25029630804362512491427684389, 10.076057426056835907991083450032, 10.77310807800828516649346093921, 11.660912226366579919961355616240, 13.18175186830275963211823994698, 13.45980597813211519577340123509, 14.44190106300493349371651260422, 14.83492933374073046373716766022, 15.98805824546179775370525476544, 16.68091984734594473788568979153, 17.056057612548230183804162014119, 18.361165709634580901986018426262, 18.483629414726505300959921259376, 19.16035604524914728981220811822, 20.43590924359151816970029160604, 21.21622517838908347685715478370

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