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Properties

Label 2-1323-63.5-c1-0-32
Degree $2$
Conductor $1323$
Sign $-0.865 - 0.500i$
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  − 1.18i·2-s + 0.593·4-s + (−1.41 − 2.45i)5-s − 3.07i·8-s + (−2.91 + 1.68i)10-s + (0.136 + 0.0789i)11-s + (−3.41 − 1.97i)13-s − 2.46·16-s + (2.07 + 3.58i)17-s + (−5.48 − 3.16i)19-s + (−0.842 − 1.45i)20-s + (0.0935 − 0.162i)22-s + (0.472 − 0.273i)23-s + (−1.52 + 2.64i)25-s + (−2.33 + 4.04i)26-s + ⋯
L(s)  = 1  − 0.838i·2-s + 0.296·4-s + (−0.634 − 1.09i)5-s − 1.08i·8-s + (−0.921 + 0.532i)10-s + (0.0412 + 0.0237i)11-s + (−0.947 − 0.546i)13-s − 0.615·16-s + (0.502 + 0.870i)17-s + (−1.25 − 0.726i)19-s + (−0.188 − 0.326i)20-s + (0.0199 − 0.0345i)22-s + (0.0986 − 0.0569i)23-s + (−0.305 + 0.528i)25-s + (−0.458 + 0.794i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.865 - 0.500i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9944239531\)
\(L(\frac12)\) \(\approx\) \(0.9944239531\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.18iT - 2T^{2} \)
5 \( 1 + (1.41 + 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.136 - 0.0789i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 + 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.07 - 3.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.48 + 3.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.472 + 0.273i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.129iT - 31T^{2} \)
37 \( 1 + (-1.23 + 2.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.99 + 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.28 - 5.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 + (-2.25 + 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.61T + 59T^{2} \)
61 \( 1 + 3.36iT - 61T^{2} \)
67 \( 1 - 1.32T + 67T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 + (13.0 - 7.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.32T + 79T^{2} \)
83 \( 1 + (3.22 + 5.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.52 + 4.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.18 - 1.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.259531498329371703595165438895, −8.423952811022982830915044213970, −7.65837298657580597347293420521, −6.77703894139827344472558263010, −5.68679647970288367264616871227, −4.63389205841190851367438283845, −3.90897565580475737989401917648, −2.79558710797190770681635895473, −1.67018184846764382362066092525, −0.38194570303074770995297128654, 2.11127697976534940574045789231, 3.02028634783110517199486720875, 4.17582414764343046113657548870, 5.26043673370514292571642193745, 6.24936888889611569954502092128, 6.90822395850782415117911546415, 7.51480956187375389741010811768, 8.102991489390138046034569435223, 9.189051425144571518908330684693, 10.16923090500079228337066759609

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