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Properties

Label 2-1323-21.20-c1-0-0
Degree $2$
Conductor $1323$
Sign $0.755 + 0.654i$
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  + 2.49i·2-s − 4.20·4-s − 1.23·5-s − 5.49i·8-s − 3.07i·10-s + 5.49i·11-s + 2.96i·13-s + 5.26·16-s − 4.31·17-s − 5.55i·19-s + 5.19·20-s − 13.6·22-s + 1.63i·23-s − 3.47·25-s − 7.39·26-s + ⋯
L(s)  = 1  + 1.76i·2-s − 2.10·4-s − 0.552·5-s − 1.94i·8-s − 0.973i·10-s + 1.65i·11-s + 0.823i·13-s + 1.31·16-s − 1.04·17-s − 1.27i·19-s + 1.16·20-s − 2.91·22-s + 0.340i·23-s − 0.694·25-s − 1.44·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1294094124\)
\(L(\frac12)\) \(\approx\) \(0.1294094124\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.49iT - 2T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 - 5.49iT - 11T^{2} \)
13 \( 1 - 2.96iT - 13T^{2} \)
17 \( 1 + 4.31T + 17T^{2} \)
19 \( 1 + 5.55iT - 19T^{2} \)
23 \( 1 - 1.63iT - 23T^{2} \)
29 \( 1 + 0.509iT - 29T^{2} \)
31 \( 1 + 8.13iT - 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + 0.354T + 41T^{2} \)
43 \( 1 + 0.0637T + 43T^{2} \)
47 \( 1 - 9.86T + 47T^{2} \)
53 \( 1 + 4.12iT - 53T^{2} \)
59 \( 1 - 3.07T + 59T^{2} \)
61 \( 1 + 6.89iT - 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 4.63iT - 71T^{2} \)
73 \( 1 - 7.03iT - 73T^{2} \)
79 \( 1 + 0.331T + 79T^{2} \)
83 \( 1 + 7.39T + 83T^{2} \)
89 \( 1 + 3.43T + 89T^{2} \)
97 \( 1 + 4.45iT - 97T^{2} \)
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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

−9.850107141843596427634615996840, −9.275486050109314900434343632453, −8.570665028208299758040986248169, −7.58519162291977704806306462969, −7.12380577919762560523672672752, −6.52642450875808572746954942078, −5.45270256414251427182906175857, −4.45161663798624276618680495695, −4.16762964364103138011385264114, −2.20951611826878483646300072927, 0.05721754742115903492735777805, 1.29686330435230562645981325753, 2.66258477719944300686322350989, 3.46829171252771643150077494416, 4.11645327309431607832105545600, 5.26794942651048475936293547611, 6.18762081163272887143016859156, 7.61379496643997599084050212438, 8.588488792841782326215823666477, 8.865867864280361779819361555282

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