Properties

Label 2-1323-63.4-c1-0-13
Degree $2$
Conductor $1323$
Sign $0.229 - 0.973i$
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  + (0.0341 − 0.0592i)2-s + (0.997 + 1.72i)4-s + 2.66·5-s + 0.273·8-s + (0.0910 − 0.157i)10-s + 1.59·11-s + (−2.62 + 4.54i)13-s + (−1.98 + 3.43i)16-s + (−3.27 + 5.67i)17-s + (−0.950 − 1.64i)19-s + (2.65 + 4.60i)20-s + (0.0546 − 0.0946i)22-s + 3.06·23-s + 2.09·25-s + (0.179 + 0.311i)26-s + ⋯
L(s)  = 1  + (0.0241 − 0.0418i)2-s + (0.498 + 0.864i)4-s + 1.19·5-s + 0.0965·8-s + (0.0287 − 0.0498i)10-s + 0.482·11-s + (−0.728 + 1.26i)13-s + (−0.496 + 0.859i)16-s + (−0.793 + 1.37i)17-s + (−0.218 − 0.377i)19-s + (0.594 + 1.02i)20-s + (0.0116 − 0.0201i)22-s + 0.639·23-s + 0.419·25-s + (0.0352 + 0.0610i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.214381766\)
\(L(\frac12)\) \(\approx\) \(2.214381766\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0341 + 0.0592i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.66T + 5T^{2} \)
11 \( 1 - 1.59T + 11T^{2} \)
13 \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.27 - 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.950 + 1.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 + (-3.19 - 5.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.35 + 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.69 + 6.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.89 + 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.09 - 1.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.406 - 0.704i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.41 - 5.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.235 - 0.407i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.597862247102819918936018414065, −9.084145537160382253875135160922, −8.298196458247840622803753782186, −7.10966937177444880960543923206, −6.64626483061798679476609826929, −5.82132317717572354552466958663, −4.58794892821645242523531308921, −3.74575362466332032262654814004, −2.38306087400394082719579492408, −1.81872521595043251057208803299, 0.886418581895892042878480400243, 2.16726578472551952058770052626, 2.91081006277613962856556477617, 4.65087361506827816322225229977, 5.38675797142927225388769635621, 6.07146033680751374314648428584, 6.85698602150961744229457648890, 7.64067666769638703835756298312, 8.991351345363295438353700811108, 9.505094061150116651520282170196

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