Properties

Label 2-1323-63.38-c1-0-34
Degree $2$
Conductor $1323$
Sign $-0.866 + 0.498i$
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  − 1.17i·2-s + 0.611·4-s + (2.16 − 3.75i)5-s − 3.07i·8-s + (−4.42 − 2.55i)10-s + (−1.87 + 1.08i)11-s + (2.25 − 1.30i)13-s − 2.40·16-s + (−0.585 + 1.01i)17-s + (2.09 − 1.20i)19-s + (1.32 − 2.29i)20-s + (1.27 + 2.20i)22-s + (−3.16 − 1.82i)23-s + (−6.88 − 11.9i)25-s + (−1.53 − 2.65i)26-s + ⋯
L(s)  = 1  − 0.833i·2-s + 0.305·4-s + (0.968 − 1.67i)5-s − 1.08i·8-s + (−1.39 − 0.807i)10-s + (−0.564 + 0.325i)11-s + (0.624 − 0.360i)13-s − 0.600·16-s + (−0.142 + 0.245i)17-s + (0.480 − 0.277i)19-s + (0.296 − 0.513i)20-s + (0.271 + 0.470i)22-s + (−0.659 − 0.380i)23-s + (−1.37 − 2.38i)25-s + (−0.300 − 0.520i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.866 + 0.498i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.866 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.218441693\)
\(L(\frac12)\) \(\approx\) \(2.218441693\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.17iT - 2T^{2} \)
5 \( 1 + (-2.16 + 3.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.87 - 1.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.25 + 1.30i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.585 - 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.589 + 0.340i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-2.55 - 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.68 - 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 + (2.79 + 1.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 + 7.17iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 + (-10.3 - 5.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.75T + 79T^{2} \)
83 \( 1 + (0.796 - 1.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.04 + 5.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.36 + 1.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.579303862472565453394457188342, −8.608850697543086146007048261313, −7.956616895435633472718968474791, −6.66187241281091827346966662480, −5.82787928111929386838715284906, −5.00058287614088481823599829779, −4.10779931041661842932992986012, −2.79212955069668639683529801324, −1.77468359271810512318162354525, −0.910739797646622335747044821396, 2.01019925707253228194278190030, 2.71326793584704798989664259151, 3.82144011550588007574592623496, 5.60093929797478544032964382186, 5.81726514140804586363153340550, 6.71958911987730667433280599261, 7.31905685404347796099014542985, 8.024986157001982306989187747431, 9.187750458761727300088374808446, 9.988552742970281738850712418417

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