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Properties

Label 2-1323-21.20-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.156 + 0.987i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67i·2-s − 5.13·4-s + 3.20·5-s + 8.37i·8-s − 8.55i·10-s + 4.06i·11-s + 2.99i·13-s + 12.0·16-s + 2.54·17-s − 0.165i·19-s − 16.4·20-s + 10.8·22-s + 4.04i·23-s + 5.26·25-s + 8.00·26-s + ⋯
L(s)  = 1  − 1.88i·2-s − 2.56·4-s + 1.43·5-s + 2.95i·8-s − 2.70i·10-s + 1.22i·11-s + 0.831i·13-s + 3.02·16-s + 0.616·17-s − 0.0379i·19-s − 3.67·20-s + 2.31·22-s + 0.843i·23-s + 1.05·25-s + 1.57·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763135246\)
\(L(\frac12)\) \(\approx\) \(1.763135246\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 2.67iT - 2T^{2} \)
5 \( 1 - 3.20T + 5T^{2} \)
11 \( 1 - 4.06iT - 11T^{2} \)
13 \( 1 - 2.99iT - 13T^{2} \)
17 \( 1 - 2.54T + 17T^{2} \)
19 \( 1 + 0.165iT - 19T^{2} \)
23 \( 1 - 4.04iT - 23T^{2} \)
29 \( 1 + 2.12iT - 29T^{2} \)
31 \( 1 - 7.70iT - 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + 0.629T + 41T^{2} \)
43 \( 1 - 3.94T + 43T^{2} \)
47 \( 1 + 5.45T + 47T^{2} \)
53 \( 1 + 2.48iT - 53T^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + 1.14T + 67T^{2} \)
71 \( 1 - 9.33iT - 71T^{2} \)
73 \( 1 + 6.10iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 3.76T + 83T^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.724583326681731535250837492874, −9.271383013904942983244364118819, −8.243385919695035663776264127156, −6.97126805046528392790761853142, −5.76687516494144161371291088986, −4.93653002888452552319981718442, −4.12158319528299235248350029533, −2.91650611465486858632038798846, −2.02554443854544269464578139935, −1.36457589187075634776528072207, 0.834972320083100621018638902163, 2.82757764577683708240378394099, 4.18373421213912666596018209716, 5.30019487451808637572094064702, 5.90889862889434803073879106192, 6.20289103669361852511290104530, 7.30766264815843300002758601965, 8.146268921553860353416428668491, 8.737335623231802020425733255300, 9.616115835023916139679908551689

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