Properties

Label 2-1323-63.47-c1-0-10
Degree $2$
Conductor $1323$
Sign $0.586 - 0.810i$
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  + (1.58 − 0.916i)2-s + (0.678 − 1.17i)4-s − 0.645·5-s + 1.17i·8-s + (−1.02 + 0.591i)10-s + 5.31i·11-s + (−4.44 + 2.56i)13-s + (2.43 + 4.22i)16-s + (−0.814 − 1.41i)17-s + (2.09 + 1.20i)19-s + (−0.437 + 0.758i)20-s + (4.86 + 8.43i)22-s + 1.47i·23-s − 4.58·25-s + (−4.69 + 8.13i)26-s + ⋯
L(s)  = 1  + (1.12 − 0.647i)2-s + (0.339 − 0.587i)4-s − 0.288·5-s + 0.416i·8-s + (−0.323 + 0.187i)10-s + 1.60i·11-s + (−1.23 + 0.711i)13-s + (0.609 + 1.05i)16-s + (−0.197 − 0.342i)17-s + (0.479 + 0.276i)19-s + (−0.0978 + 0.169i)20-s + (1.03 + 1.79i)22-s + 0.306i·23-s − 0.916·25-s + (−0.921 + 1.59i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.205229479\)
\(L(\frac12)\) \(\approx\) \(2.205229479\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.916i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.645T + 5T^{2} \)
11 \( 1 - 5.31iT - 11T^{2} \)
13 \( 1 + (4.44 - 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.814 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (-6.43 - 3.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.90 + 2.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.99 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.99 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.54 + 2.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.04 + 1.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.47 + 2.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.18 - 5.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.07 + 8.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.48 + 6.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.51 + 6.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.16 - 3.74i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.3 - 8.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.764813838976989454320192330613, −9.267988534659213789227133917552, −7.86404509479246011778349722441, −7.35335954963619549618456896279, −6.30105374227306137794621739929, −5.09947302148130020370559209897, −4.61187662861249494428906349538, −3.82437888321356488602232603295, −2.63196690779138408841991751808, −1.84769301587960748345738648170, 0.61136957920628583842454825766, 2.71316804261990636781724552013, 3.56290387687441159699779621114, 4.49600606946065819879630362189, 5.40494154692435024639671947978, 5.96857873706415108610391055369, 6.88205239173233695126662980741, 7.71990249471267731130531992104, 8.446047729847793227300913303373, 9.524711116820059824801592423696

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