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Properties

Label 2-1323-63.47-c1-0-0
Degree $2$
Conductor $1323$
Sign $-0.967 + 0.252i$
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  + (0.621 − 0.359i)2-s + (−0.742 + 1.28i)4-s − 1.44·5-s + 2.50i·8-s + (−0.900 + 0.519i)10-s + 1.80i·11-s + (−1.88 + 1.09i)13-s + (−0.585 − 1.01i)16-s + (−1.95 − 3.38i)17-s + (3.47 + 2.00i)19-s + (1.07 − 1.86i)20-s + (0.646 + 1.11i)22-s − 5.67i·23-s − 2.90·25-s + (−0.783 + 1.35i)26-s + ⋯
L(s)  = 1  + (0.439 − 0.253i)2-s + (−0.371 + 0.642i)4-s − 0.647·5-s + 0.884i·8-s + (−0.284 + 0.164i)10-s + 0.542i·11-s + (−0.523 + 0.302i)13-s + (−0.146 − 0.253i)16-s + (−0.473 − 0.820i)17-s + (0.797 + 0.460i)19-s + (0.240 − 0.416i)20-s + (0.137 + 0.238i)22-s − 1.18i·23-s − 0.580·25-s + (−0.153 + 0.266i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.967 + 0.252i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06539806268\)
\(L(\frac12)\) \(\approx\) \(0.06539806268\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.621 + 0.359i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 - 1.80iT - 11T^{2} \)
13 \( 1 + (1.88 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.47 - 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.67iT - 23T^{2} \)
29 \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.45 - 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.90 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.996 + 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 - 8.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.03 - 1.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.156 + 0.270i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (2.42 - 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.21 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.60 - 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.28 + 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.4 + 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.987586918630237620741559513005, −9.239615004903200597575043979283, −8.401175277542356995187330297034, −7.57823338962348509167587797678, −7.06386493297764624845003727241, −5.71452086846812075745827643051, −4.70592600518773423796662825964, −4.14768794730166123687516306546, −3.15389648893908541530119663196, −2.12624489124918355561848030124, 0.02307155829891234463264964032, 1.57401992923465864789901895335, 3.28635687399598336689241848689, 4.01421053798268240831191230237, 5.08372483664884513591556391079, 5.66651045051555991135134821281, 6.66206970884714944986035885909, 7.49276628589220443112925126518, 8.329936954242218825552114326120, 9.296332303966955108207498613672

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