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Properties

Label 2-1323-63.5-c1-0-29
Degree $2$
Conductor $1323$
Sign $-0.689 + 0.724i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.689 + 0.724i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.423855543\)
\(L(\frac12)\) \(\approx\) \(1.423855543\)
\(L(\frac{3}{2})\) 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

−9.703434293585290916152518780813, −8.474713358391276742699763137857, −7.59799147897224101264868372381, −6.97927020786807312814878257480, −6.04235053750619649524925025793, −5.07989656848377385709409445363, −4.04451847438257394378626298777, −2.74968659468875170524103525419, −2.27765464601414338460326511091, −0.52528022959808143256569187626, 1.83719884773199438189027837458, 2.60155411107051664429609370939, 4.01566303338577078387288956856, 4.97480115438411420656271117693, 5.95240482926212666322805766958, 6.74375939419026140909535165314, 7.31849872098920740380488069470, 8.246115329401785236711885859624, 8.915235749595167473728756984842, 10.12266431108900387962645223336

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