Properties

Label 2-1323-63.4-c1-0-34
Degree $2$
Conductor $1323$
Sign $-0.841 - 0.540i$
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  + (1.26 − 2.19i)2-s + (−2.20 − 3.82i)4-s + 0.879·5-s − 6.10·8-s + (1.11 − 1.92i)10-s − 3.87·11-s + (2.72 − 4.72i)13-s + (−3.31 + 5.74i)16-s + (0.826 − 1.43i)17-s + (−1.20 − 2.08i)19-s + (−1.93 − 3.35i)20-s + (−4.91 + 8.50i)22-s − 3.16·23-s − 4.22·25-s + (−6.90 − 11.9i)26-s + ⋯
L(s)  = 1  + (0.895 − 1.55i)2-s + (−1.10 − 1.91i)4-s + 0.393·5-s − 2.15·8-s + (0.352 − 0.609i)10-s − 1.16·11-s + (0.756 − 1.30i)13-s + (−0.829 + 1.43i)16-s + (0.200 − 0.347i)17-s + (−0.276 − 0.479i)19-s + (−0.433 − 0.751i)20-s + (−1.04 + 1.81i)22-s − 0.659·23-s − 0.845·25-s + (−1.35 − 2.34i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041485598\)
\(L(\frac12)\) \(\approx\) \(2.041485598\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 + 2.19i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.879T + 5T^{2} \)
11 \( 1 + 3.87T + 11T^{2} \)
13 \( 1 + (-2.72 + 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.826 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 + (3.02 + 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.27 - 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.27 - 3.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.592 - 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.511 - 0.885i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.64 + 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.33 - 5.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.29 + 2.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + (-6.39 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.51 - 9.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.25 + 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.701976158347184535743614750533, −8.498074933491676099193684111356, −7.68772688330196845980201925798, −6.16405870985474125118705787054, −5.50880473137366003554907900470, −4.78477852443994247448670642444, −3.68720964143261285185181307136, −2.86257724704403545550713655087, −2.02375430070872273006404283559, −0.60298990898763842632623615899, 2.10456051196958451822220517683, 3.62025655191075065478596063312, 4.30565867035987680399091601771, 5.35780130376155986254672632307, 5.93981238472016407153613239478, 6.64534170906985277683248946387, 7.55728708544042069489097865681, 8.184195367009296707442025975107, 8.967523105480884549713258321019, 9.902375313318516075809650248282

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