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Properties

Label 2-1323-9.4-c1-0-9
Degree $2$
Conductor $1323$
Sign $-0.949 - 0.313i$
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  + (0.649 + 1.12i)2-s + (0.155 − 0.268i)4-s + (−1.76 + 3.05i)5-s + 3.00·8-s − 4.58·10-s + (0.589 + 1.02i)11-s + (−1.61 + 2.78i)13-s + (1.64 + 2.84i)16-s − 4.90·17-s + 6.86·19-s + (0.547 + 0.947i)20-s + (−0.765 + 1.32i)22-s + (−2.14 + 3.72i)23-s + (−3.71 − 6.43i)25-s − 4.18·26-s + ⋯
L(s)  = 1  + (0.459 + 0.796i)2-s + (0.0775 − 0.134i)4-s + (−0.788 + 1.36i)5-s + 1.06·8-s − 1.44·10-s + (0.177 + 0.307i)11-s + (−0.446 + 0.773i)13-s + (0.410 + 0.710i)16-s − 1.18·17-s + 1.57·19-s + (0.122 + 0.211i)20-s + (−0.163 + 0.282i)22-s + (−0.448 + 0.776i)23-s + (−0.743 − 1.28i)25-s − 0.821·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.949 - 0.313i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.949 - 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628952428\)
\(L(\frac12)\) \(\approx\) \(1.628952428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 - 1.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.589 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.61 - 2.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.90T + 17T^{2} \)
19 \( 1 - 6.86T + 19T^{2} \)
23 \( 1 + (2.14 - 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.36 + 2.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.960 - 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (3.32 - 5.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.316 + 0.548i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.22T + 53T^{2} \)
59 \( 1 + (4.10 - 7.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.82 + 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 + (0.502 + 0.869i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.65 + 6.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-5.46 - 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.04248230917984343632337756939, −9.234696642976387651331742351544, −7.905080881451432235167879514306, −7.29222202267987442911700903065, −6.82665739384663087360744601564, −6.06923264234261803384020697351, −4.96797093646204785781449613754, −4.12038376014758758643141516178, −3.11865878938728904584340336239, −1.83723047840157232710888325960, 0.56607163934003673705860263090, 1.90605238492996864429125127877, 3.19873384854631647402049415027, 3.99746413809041540733367581570, 4.82813850720808586984026203807, 5.49452773934978727640927936796, 7.02798172375672081847004055454, 7.71680832325151389286358407576, 8.530768457563252092439250013790, 9.157888749442098952799331608667

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