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Properties

Label 2-1323-63.38-c1-0-16
Degree $2$
Conductor $1323$
Sign $0.958 - 0.286i$
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  − 0.424i·2-s + 1.82·4-s + (−1.80 + 3.12i)5-s − 1.62i·8-s + (1.32 + 0.765i)10-s + (3.20 − 1.85i)11-s + (5.23 − 3.02i)13-s + 2.95·16-s + (−0.532 + 0.921i)17-s + (−3.16 + 1.82i)19-s + (−3.28 + 5.68i)20-s + (−0.786 − 1.36i)22-s + (0.314 + 0.181i)23-s + (−4.00 − 6.94i)25-s + (−1.28 − 2.22i)26-s + ⋯
L(s)  = 1  − 0.299i·2-s + 0.910·4-s + (−0.806 + 1.39i)5-s − 0.572i·8-s + (0.419 + 0.241i)10-s + (0.967 − 0.558i)11-s + (1.45 − 0.838i)13-s + 0.738·16-s + (−0.129 + 0.223i)17-s + (−0.725 + 0.418i)19-s + (−0.734 + 1.27i)20-s + (−0.167 − 0.290i)22-s + (0.0655 + 0.0378i)23-s + (−0.801 − 1.38i)25-s + (−0.251 − 0.435i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.958 - 0.286i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.035098820\)
\(L(\frac12)\) \(\approx\) \(2.035098820\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.424iT - 2T^{2} \)
5 \( 1 + (1.80 - 3.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.20 + 1.85i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.23 + 3.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.532 - 0.921i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.16 - 1.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.314 - 0.181i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.857 - 0.495i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.08iT - 31T^{2} \)
37 \( 1 + (-4.00 - 6.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.09 - 3.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.89 - 3.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.67T + 47T^{2} \)
53 \( 1 + (-3.92 - 2.26i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 0.0275iT - 61T^{2} \)
67 \( 1 + 9.72T + 67T^{2} \)
71 \( 1 - 5.55iT - 71T^{2} \)
73 \( 1 + (1.95 + 1.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + (1.52 - 2.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.47 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.67 - 0.964i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.07156011614314814153136046748, −8.674541912476081816666154012965, −8.002154805678518554965322869151, −7.11965168555418276134104191832, −6.35798192259198434380074540755, −5.96473220788226621628278367302, −4.05858852251971402699761505001, −3.43960081043050278299683542620, −2.70257736905925507442415966760, −1.22664103273937169779491568785, 1.03702754830664144292098094748, 2.11033797713721966381208414642, 3.80805826302105603323974548711, 4.32858707373806815548119009459, 5.47224680279294463431340483276, 6.39887588366140768930076613998, 7.10880282651361751213244143808, 8.005793012652012470156199047569, 8.823229580204319249769177435647, 9.177051800420319206145958216650

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