Properties

Label 2-1323-63.20-c1-0-20
Degree $2$
Conductor $1323$
Sign $-0.639 - 0.769i$
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  + (2.34 + 1.35i)2-s + (2.66 + 4.62i)4-s + (0.601 + 1.04i)5-s + 9.04i·8-s + 3.25i·10-s + (2.15 + 1.24i)11-s + (−1.63 + 0.942i)13-s + (−6.90 + 11.9i)16-s − 1.20·17-s − 7.47i·19-s + (−3.21 + 5.56i)20-s + (3.36 + 5.83i)22-s + (−2.63 + 1.52i)23-s + (1.77 − 3.07i)25-s − 5.10·26-s + ⋯
L(s)  = 1  + (1.65 + 0.957i)2-s + (1.33 + 2.31i)4-s + (0.268 + 0.465i)5-s + 3.19i·8-s + 1.03i·10-s + (0.649 + 0.374i)11-s + (−0.452 + 0.261i)13-s + (−1.72 + 2.99i)16-s − 0.291·17-s − 1.71i·19-s + (−0.717 + 1.24i)20-s + (0.718 + 1.24i)22-s + (−0.549 + 0.317i)23-s + (0.355 − 0.615i)25-s − 1.00·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.639 - 0.769i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.639 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.553216770\)
\(L(\frac12)\) \(\approx\) \(4.553216770\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 - 1.35i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.601 - 1.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.15 - 1.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.63 - 0.942i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 + 7.47iT - 19T^{2} \)
23 \( 1 + (2.63 - 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.173 - 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.03 + 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (-3.36 - 5.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00656 + 0.0113i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.717 - 1.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.90iT - 53T^{2} \)
59 \( 1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.73 + 5.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 - 8.67iT - 73T^{2} \)
79 \( 1 + (2.74 - 4.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.898334876820853314164442843229, −8.888702734537659803494087959646, −7.916165083046737492277272295679, −7.00263202076529394394790156416, −6.62701926537343875060125284426, −5.80844871378234335920908633087, −4.74616698468150152516916466587, −4.26983336921941493194141047716, −3.06440355699952621983757396858, −2.27176495335805114221266386197, 1.17851437973093011941503975583, 2.18104784815160844737306436126, 3.32508297607783526836454527923, 4.09553314496660936865983858101, 4.94347945408081263564602271910, 5.79828202140015294137756412582, 6.32806694313896293422728428889, 7.47168156287287526832996906497, 8.738131094216270727812353680532, 9.730123724845301721209831998786

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