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Properties

Label 1-1323-1323.25-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.0261 - 0.999i$
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  + (−0.969 + 0.246i)2-s + (0.878 − 0.478i)4-s + (0.456 − 0.889i)5-s + (−0.733 + 0.680i)8-s + (−0.222 + 0.974i)10-s + (−0.969 + 0.246i)11-s + (0.995 + 0.0995i)13-s + (0.542 − 0.840i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.0249 − 0.999i)20-s + (0.878 − 0.478i)22-s + (−0.853 − 0.521i)23-s + (−0.583 − 0.811i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.969 + 0.246i)2-s + (0.878 − 0.478i)4-s + (0.456 − 0.889i)5-s + (−0.733 + 0.680i)8-s + (−0.222 + 0.974i)10-s + (−0.969 + 0.246i)11-s + (0.995 + 0.0995i)13-s + (0.542 − 0.840i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.0249 − 0.999i)20-s + (0.878 − 0.478i)22-s + (−0.853 − 0.521i)23-s + (−0.583 − 0.811i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0261 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0261 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0261 - 0.999i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.0261 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6180528771 - 0.6344113493i\)
\(L(\frac12)\) \(\approx\) \(0.6180528771 - 0.6344113493i\)
\(L(1)\) \(\approx\) \(0.7260524173 - 0.1545394106i\)
\(L(1)\) \(\approx\) \(0.7260524173 - 0.1545394106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.969 + 0.246i)T \)
5 \( 1 + (0.456 - 0.889i)T \)
11 \( 1 + (-0.969 + 0.246i)T \)
13 \( 1 + (0.995 + 0.0995i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.853 - 0.521i)T \)
29 \( 1 + (-0.853 + 0.521i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.542 + 0.840i)T \)
43 \( 1 + (-0.998 - 0.0498i)T \)
47 \( 1 + (0.270 - 0.962i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.124 - 0.992i)T \)
61 \( 1 + (0.878 + 0.478i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.988 + 0.149i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.995 - 0.0995i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−21.05164094092043248900267678021, −20.39927549233722192644033350348, −19.36694743138537046805266547331, −18.745220419863288981331180132107, −18.14151619013191796339069258826, −17.622910395562430894147669853657, −16.68954152006629342928994399646, −15.76710809669924534631572594085, −15.37162993096275911870305780781, −14.174716200989733254249209960204, −13.481831180468567362461332291958, −12.51173084822042963913856887472, −11.57306313572206112535294368237, −10.8100777017506038396429782699, −10.281324868743603699935684533342, −9.58028139876046357789255678385, −8.553659590610361937752036796147, −7.79606031238805484169325733086, −7.108635763159277016349021860128, −6.04876420487066667796966784402, −5.538971837082019826028325794927, −3.70668356185269085178086248217, −3.1471928468700210637521295653, −2.132754261768826355337025936917, −1.23125816531782170620959238772, 0.50150944795924066777379172681, 1.54032882045845363843831581374, 2.41305756720341225054837246313, 3.60680433518765415521846148405, 5.074658707459693185426895854904, 5.55175252212748015854893540134, 6.50361469540257960094326406535, 7.57106261605551477003398533817, 8.18553406022182793352484910111, 8.98499152577053916146934076284, 9.769691638917517633938456738829, 10.31640835083687080431903116927, 11.39493630526906803820381673737, 12.06554561600707816936004902445, 13.09549640080395714895902317662, 13.79957564766979941466107853591, 14.77556419294590912602828329586, 15.84959677572635661495655368717, 16.20252483568745424732861711790, 16.849631955075194257419129537093, 17.97582560697576937111809588831, 18.1938401911833595472044044003, 19.034505786032064840897439683668, 20.19321934736441259679520426864, 20.59322411483373214047245932223

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