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Properties

Label 1-1323-1323.1033-r0-0-0
Degree $1$
Conductor $1323$
Sign $-0.523 + 0.852i$
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  + (0.878 + 0.478i)2-s + (0.542 + 0.840i)4-s + (−0.583 + 0.811i)5-s + (0.0747 + 0.997i)8-s + (−0.900 + 0.433i)10-s + (0.878 + 0.478i)11-s + (0.980 − 0.198i)13-s + (−0.411 + 0.911i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.998 − 0.0498i)20-s + (0.542 + 0.840i)22-s + (0.456 − 0.889i)23-s + (−0.318 − 0.947i)25-s + (0.955 + 0.294i)26-s + ⋯
L(s)  = 1  + (0.878 + 0.478i)2-s + (0.542 + 0.840i)4-s + (−0.583 + 0.811i)5-s + (0.0747 + 0.997i)8-s + (−0.900 + 0.433i)10-s + (0.878 + 0.478i)11-s + (0.980 − 0.198i)13-s + (−0.411 + 0.911i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.998 − 0.0498i)20-s + (0.542 + 0.840i)22-s + (0.456 − 0.889i)23-s + (−0.318 − 0.947i)25-s + (0.955 + 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.523 + 0.852i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ -0.523 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299264589 + 2.323076023i\)
\(L(\frac12)\) \(\approx\) \(1.299264589 + 2.323076023i\)
\(L(1)\) \(\approx\) \(1.452874514 + 0.9940123178i\)
\(L(1)\) \(\approx\) \(1.452874514 + 0.9940123178i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.878 + 0.478i)T \)
5 \( 1 + (-0.583 + 0.811i)T \)
11 \( 1 + (0.878 + 0.478i)T \)
13 \( 1 + (0.980 - 0.198i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.456 - 0.889i)T \)
29 \( 1 + (0.456 + 0.889i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.411 - 0.911i)T \)
43 \( 1 + (0.995 - 0.0995i)T \)
47 \( 1 + (-0.853 + 0.521i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (-0.969 - 0.246i)T \)
61 \( 1 + (0.542 - 0.840i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.955 + 0.294i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.980 + 0.198i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−20.61181036967795570641858148898, −20.13292955006492397057217368161, −19.38563003559423054883717034955, −18.69840344788765163685034325453, −17.702531336885121104099206042148, −16.430746894921146844591930196265, −16.14923687606278446586701646345, −15.29933823682258507594172160297, −14.43143364863253302900748588082, −13.48139331974305780499613875414, −13.211000385530071184013723809798, −11.97756896162288819282378814364, −11.570177009230663575851651024797, −11.03149604327749355661941316364, −9.60169919259174128256526695537, −9.19224986932307119667344574420, −8.04040796826689036902778815954, −7.08954083312316488173989743235, −6.125967545511689262168766154993, −5.331727665113133328916484930329, −4.46221329624095859187547647457, −3.719893213998937109451110736227, −2.97388678007643141820117389284, −1.54239021652238244141060445920, −0.84653840759643992911703563789, 1.47614865295158866752402179568, 2.712151040228006260099619946989, 3.593808024095080247862306842358, 4.11218655219735056865673505229, 5.187804056137017696654386540285, 6.28127154199427195346501012362, 6.72322940913980568169548372740, 7.612013393846466060983366100226, 8.376259299227893192725411899143, 9.34217387458175797622870202153, 10.851044340265704449865366111677, 10.99275866525722360919029521926, 12.18022359949055873666382372212, 12.61925881915373298169184862453, 13.72224678541429247874828220182, 14.434355973340944413979395532703, 14.92763734814563697827296924298, 15.77574279777418928157812726681, 16.33107683018577411846398449780, 17.34160635566419310668881268609, 18.0501607247066610064203884714, 18.9080601277400544828013988244, 19.9744844001639512669557711496, 20.34014256049956928891811555833, 21.47166713236064778608560990061

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