Properties

Label 2-1323-63.20-c1-0-22
Degree $2$
Conductor $1323$
Sign $0.997 - 0.0711i$
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  + (1.28 + 0.742i)2-s + (0.101 + 0.176i)4-s + (0.154 + 0.267i)5-s − 2.66i·8-s + 0.457i·10-s + (2.73 + 1.58i)11-s + (3.00 − 1.73i)13-s + (2.18 − 3.78i)16-s − 4.88·17-s + 5.34i·19-s + (−0.0314 + 0.0544i)20-s + (2.34 + 4.06i)22-s + (5.17 − 2.98i)23-s + (2.45 − 4.24i)25-s + 5.14·26-s + ⋯
L(s)  = 1  + (0.909 + 0.524i)2-s + (0.0509 + 0.0882i)4-s + (0.0689 + 0.119i)5-s − 0.942i·8-s + 0.144i·10-s + (0.825 + 0.476i)11-s + (0.833 − 0.481i)13-s + (0.545 − 0.945i)16-s − 1.18·17-s + 1.22i·19-s + (−0.00702 + 0.0121i)20-s + (0.500 + 0.866i)22-s + (1.07 − 0.622i)23-s + (0.490 − 0.849i)25-s + 1.00·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.997 - 0.0711i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.997 - 0.0711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.790328240\)
\(L(\frac12)\) \(\approx\) \(2.790328240\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 0.742i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.154 - 0.267i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.73 - 1.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.00 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 - 5.34iT - 19T^{2} \)
23 \( 1 + (-5.17 + 2.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.70 + 1.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.51 + 3.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + (2.58 + 4.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.75 + 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.23 - 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.0855iT - 53T^{2} \)
59 \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 2.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0554 - 0.0959i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 - 9.61iT - 73T^{2} \)
79 \( 1 + (2.56 - 4.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.42 + 7.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + (10.9 + 6.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.645849625771257705435359242515, −8.809901725411669356602093074392, −7.897399609819881662385214682371, −6.77573998858460175506384297839, −6.33605253164999131398503911296, −5.53968944452511156787121223930, −4.39701246756665840700673897048, −3.98357182677420587213708473042, −2.64808836651116756696966259793, −1.05385588077197252328203711783, 1.33250906865491664311445676120, 2.68609604699733741245498921771, 3.53225926515981048766669274375, 4.45734866019692630141324637968, 5.10608072901343104826319070617, 6.25809189547619158961003337762, 6.90791030329783323732430590654, 8.168845859313087149095908230758, 8.944466982598516556967769293866, 9.399022918155320259537654967489

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