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Properties

Label 2-1323-63.38-c1-0-22
Degree $2$
Conductor $1323$
Sign $0.879 - 0.475i$
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  + 0.664i·2-s + 1.55·4-s + (0.0141 − 0.0245i)5-s + 2.36i·8-s + (0.0162 + 0.00940i)10-s + (−0.885 + 0.511i)11-s + (4.87 − 2.81i)13-s + 1.54·16-s + (2.83 − 4.91i)17-s + (1.81 − 1.04i)19-s + (0.0220 − 0.0382i)20-s + (−0.339 − 0.588i)22-s + (−6.28 − 3.63i)23-s + (2.49 + 4.32i)25-s + (1.87 + 3.24i)26-s + ⋯
L(s)  = 1  + 0.469i·2-s + 0.779·4-s + (0.00632 − 0.0109i)5-s + 0.835i·8-s + (0.00514 + 0.00297i)10-s + (−0.266 + 0.154i)11-s + (1.35 − 0.781i)13-s + 0.386·16-s + (0.688 − 1.19i)17-s + (0.415 − 0.240i)19-s + (0.00493 − 0.00854i)20-s + (−0.0723 − 0.125i)22-s + (−1.31 − 0.757i)23-s + (0.499 + 0.865i)25-s + (0.366 + 0.635i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.233284882\)
\(L(\frac12)\) \(\approx\) \(2.233284882\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.664iT - 2T^{2} \)
5 \( 1 + (-0.0141 + 0.0245i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.885 - 0.511i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.87 + 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.81 + 1.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.28 + 3.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.52 - 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.31iT - 31T^{2} \)
37 \( 1 + (-1.23 - 2.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + (-10.0 - 5.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.02T + 59T^{2} \)
61 \( 1 - 2.36iT - 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 7.93iT - 71T^{2} \)
73 \( 1 + (9.43 + 5.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (-3.07 + 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.77 + 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.786928527835856884276713973783, −8.636137658325023994091966022130, −7.971312034625216878323174326999, −7.28399347008325771789058013327, −6.36414692378336599070651712708, −5.70207742531650268880667843113, −4.82949305535998508599721700415, −3.41801280433997546886724219804, −2.60568604096923525837083099268, −1.15497738635425586040232199775, 1.23330671486937590877083147450, 2.19535006250808285987518491222, 3.47617647196964963535126856783, 4.04400048759674355125570557647, 5.60685744890620082913273782974, 6.23284414264604065814837217256, 7.02515778518989695801754154496, 8.081053497781325784186296505858, 8.633953002702943729655225725484, 9.950855451827524118451374720647

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