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Properties

Label 2-1323-63.47-c1-0-8
Degree $2$
Conductor $1323$
Sign $0.517 - 0.855i$
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  + (1.80 − 1.04i)2-s + (1.17 − 2.03i)4-s − 3.30·5-s − 0.717i·8-s + (−5.96 + 3.44i)10-s + 2.66i·11-s + (−2.11 + 1.21i)13-s + (1.59 + 2.76i)16-s + (3.59 + 6.21i)17-s + (−4.24 − 2.45i)19-s + (−3.87 + 6.70i)20-s + (2.77 + 4.80i)22-s + 4.99i·23-s + 5.92·25-s + (−2.54 + 4.40i)26-s + ⋯
L(s)  = 1  + (1.27 − 0.736i)2-s + (0.586 − 1.01i)4-s − 1.47·5-s − 0.253i·8-s + (−1.88 + 1.08i)10-s + 0.802i·11-s + (−0.585 + 0.338i)13-s + (0.399 + 0.691i)16-s + (0.870 + 1.50i)17-s + (−0.974 − 0.562i)19-s + (−0.866 + 1.50i)20-s + (0.591 + 1.02i)22-s + 1.04i·23-s + 1.18·25-s + (−0.498 + 0.863i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.517 - 0.855i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.517 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.667059902\)
\(L(\frac12)\) \(\approx\) \(1.667059902\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
11 \( 1 - 2.66iT - 11T^{2} \)
13 \( 1 + (2.11 - 1.21i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.59 - 6.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.24 + 2.45i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.99iT - 23T^{2} \)
29 \( 1 + (5.50 + 3.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.30 - 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.844 + 1.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.553 - 0.958i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.94 - 5.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.44 + 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.07iT - 71T^{2} \)
73 \( 1 + (6.94 - 4.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.04 + 1.80i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.541 - 0.937i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.47 - 5.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.10015190164977317682584733445, −8.947378866895105267515255251734, −7.937178242804407534837785201802, −7.40226679991276116518209183188, −6.23950905109451268177861480355, −5.25623573489713024799809013160, −4.18203318287336636228573665863, −4.02564177229834381373716371536, −2.89861973260399397172865633556, −1.71384730309356607449697440672, 0.44526219743628979839436063190, 2.88013347294150025723283334896, 3.59636083838643480805996069081, 4.45946715342845450292034273048, 5.15190706663717369216180174965, 6.10317198784170592922856765171, 6.99838670497192349940811650848, 7.69801089874474259126089627320, 8.243345969565727598509513342295, 9.388826085365175278903086852162

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