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Properties

Label 2-1323-63.5-c1-0-7
Degree $2$
Conductor $1323$
Sign $0.908 + 0.417i$
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  − 2.08i·2-s − 2.34·4-s + (1.65 + 2.86i)5-s + 0.717i·8-s + (5.96 − 3.44i)10-s + (2.30 + 1.33i)11-s + (−2.11 − 1.21i)13-s − 3.19·16-s + (3.59 + 6.21i)17-s + (4.24 + 2.45i)19-s + (−3.87 − 6.70i)20-s + (2.77 − 4.80i)22-s + (−4.32 + 2.49i)23-s + (−2.96 + 5.12i)25-s + (−2.54 + 4.40i)26-s + ⋯
L(s)  = 1  − 1.47i·2-s − 1.17·4-s + (0.738 + 1.27i)5-s + 0.253i·8-s + (1.88 − 1.08i)10-s + (0.694 + 0.401i)11-s + (−0.585 − 0.338i)13-s − 0.798·16-s + (0.870 + 1.50i)17-s + (0.974 + 0.562i)19-s + (−0.866 − 1.50i)20-s + (0.591 − 1.02i)22-s + (−0.901 + 0.520i)23-s + (−0.592 + 1.02i)25-s + (−0.498 + 0.863i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.790451729\)
\(L(\frac12)\) \(\approx\) \(1.790451729\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.08iT - 2T^{2} \)
5 \( 1 + (-1.65 - 2.86i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.30 - 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.11 + 1.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.59 - 6.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.32 - 2.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.50 - 3.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.66iT - 31T^{2} \)
37 \( 1 + (-0.844 + 1.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.553 + 0.958i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.88T + 47T^{2} \)
53 \( 1 + (-8.94 + 5.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 + 5.13iT - 61T^{2} \)
67 \( 1 - 8.33T + 67T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + (-6.94 + 4.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 + (-1.04 - 1.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.541 - 0.937i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.47 + 5.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.889951862171307166969803789967, −9.339108042475403878654876618639, −8.009455658504080998620779730618, −7.09673144918264289326541526339, −6.22481453043599695362964909473, −5.32791359873970462678609983518, −3.82427220413243381118583209114, −3.36627272759339901332915482440, −2.24016379437728228644861871147, −1.48154558981336014020474987391, 0.78424642002971772145851892237, 2.35808534872438115897804397363, 4.06895196645370926005841428640, 5.10009607947125980129253435181, 5.45432768213136967019294382731, 6.34170436690300313095517761887, 7.26504845188230668883081413460, 7.916731774698168092489741049406, 8.893965103907463016479055644574, 9.341081463434799375161258967275

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