Properties

Label 2-1323-63.41-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: $\chi_{1323} (881, \cdot )$
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

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

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