Properties

Label 2-1323-63.20-c1-0-30
Degree $2$
Conductor $1323$
Sign $0.456 + 0.889i$
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  + (0.850 + 0.490i)2-s + (−0.518 − 0.897i)4-s + (0.940 + 1.62i)5-s − 2.98i·8-s + 1.84i·10-s + (−3.54 − 2.04i)11-s + (3.51 − 2.02i)13-s + (0.426 − 0.738i)16-s − 1.62·17-s − 8.12i·19-s + (0.974 − 1.68i)20-s + (−2.00 − 3.47i)22-s + (−3.73 + 2.15i)23-s + (0.730 − 1.26i)25-s + 3.98·26-s + ⋯
L(s)  = 1  + (0.601 + 0.347i)2-s + (−0.259 − 0.448i)4-s + (0.420 + 0.728i)5-s − 1.05i·8-s + 0.583i·10-s + (−1.06 − 0.616i)11-s + (0.974 − 0.562i)13-s + (0.106 − 0.184i)16-s − 0.393·17-s − 1.86i·19-s + (0.217 − 0.377i)20-s + (−0.427 − 0.740i)22-s + (−0.778 + 0.449i)23-s + (0.146 − 0.253i)25-s + 0.781·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.861319873\)
\(L(\frac12)\) \(\approx\) \(1.861319873\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.850 - 0.490i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.940 - 1.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.54 + 2.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.51 + 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 + 8.12iT - 19T^{2} \)
23 \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.542 - 0.313i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.69 + 2.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.94T + 37T^{2} \)
41 \( 1 + (0.912 + 1.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.96 + 6.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.37iT - 53T^{2} \)
59 \( 1 + (-4.08 - 7.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.26 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + 3.78iT - 73T^{2} \)
79 \( 1 + (4.18 - 7.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.38 - 7.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.574480109996616671624514723269, −8.679867974430608025212508551090, −7.77426735024088981273593004990, −6.68726887006373945158380578894, −6.15767623428556399700041257026, −5.37502774057023896143205398816, −4.53132128886367088360417419308, −3.36807249314810687523465395873, −2.43657280366187950807015411405, −0.64008932620281021315592230892, 1.59918747077457536338050635315, 2.65455749229297593965867839191, 3.88947997666585239716931506641, 4.52848628362361331691392291397, 5.45344240017695379588393380309, 6.17104068699011591186105336115, 7.51666709212952219248077556157, 8.272094281150335131601343155450, 8.818440510601184427560014742056, 9.841693849445907805326183373612

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