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Properties

Label 2-1323-63.25-c1-0-13
Degree $2$
Conductor $1323$
Sign $0.888 - 0.458i$
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  − 2-s − 4-s + (0.5 − 0.866i)5-s + 3·8-s + (−0.5 + 0.866i)10-s + (2.5 + 4.33i)11-s + (−2.5 − 4.33i)13-s − 16-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + (−2.5 − 4.33i)22-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s + (2.5 + 4.33i)26-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s + (0.223 − 0.387i)5-s + 1.06·8-s + (−0.158 + 0.273i)10-s + (0.753 + 1.30i)11-s + (−0.693 − 1.20i)13-s − 0.250·16-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + (−0.111 + 0.193i)20-s + (−0.533 − 0.923i)22-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s + (0.490 + 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9498928835\)
\(L(\frac12)\) \(\approx\) \(0.9498928835\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 2T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.535338170024283781412128220456, −9.072761087674040346065979525254, −8.134515448776171351797896993779, −7.48441836003620093879257282836, −6.58699393019015893816785816125, −5.30896409065956402084594706523, −4.70029101271739953483134895144, −3.70992631059854340631228662438, −2.16190965338140163366562467892, −0.980369902927552590734704936391, 0.68306832566709173924618476460, 2.09978212722409360862569811088, 3.44407408799182771456716684405, 4.43129392912592130279411663122, 5.34319958937089259603435096710, 6.52643661197486758932776076302, 7.10730281059877771206481189488, 8.180150274112877491943119789699, 8.893537615000344072015385619258, 9.452751642132007660349974921532

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