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Properties

Label 2-1323-63.58-c1-0-4
Degree $2$
Conductor $1323$
Sign $-0.823 - 0.567i$
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  + 0.495·2-s − 1.75·4-s + (1.84 + 3.19i)5-s − 1.86·8-s + (0.915 + 1.58i)10-s + (−0.446 + 0.772i)11-s + (−0.598 + 1.03i)13-s + 2.58·16-s + (−0.124 − 0.216i)17-s + (−1.40 + 2.43i)19-s + (−3.23 − 5.60i)20-s + (−0.221 + 0.383i)22-s + (1.23 + 2.14i)23-s + (−4.31 + 7.47i)25-s + (−0.296 + 0.513i)26-s + ⋯
L(s)  = 1  + 0.350·2-s − 0.877·4-s + (0.825 + 1.43i)5-s − 0.658·8-s + (0.289 + 0.501i)10-s + (−0.134 + 0.233i)11-s + (−0.165 + 0.287i)13-s + 0.646·16-s + (−0.0303 − 0.0525i)17-s + (−0.322 + 0.557i)19-s + (−0.724 − 1.25i)20-s + (−0.0471 + 0.0817i)22-s + (0.258 + 0.447i)23-s + (−0.863 + 1.49i)25-s + (−0.0581 + 0.100i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141326240\)
\(L(\frac12)\) \(\approx\) \(1.141326240\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.495T + 2T^{2} \)
5 \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.446 - 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.124 + 0.216i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (-4.94 - 8.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (-0.915 - 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.79T + 79T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.52 + 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.927465405155192855927918455630, −9.381741995679688170384385847220, −8.377794587695502165988397699886, −7.38890732056041464354973074170, −6.54477432377138314385102710777, −5.81216616578582611195566188523, −4.97018365939435592521828516286, −3.82086646328885056032683619006, −3.01275337666968186129323919894, −1.86508666528998813296747110855, 0.41840108015786175268440157657, 1.75959885512845895068118612891, 3.24471547143822943871512285588, 4.40610355434559608593970986441, 5.09975886595183271968921847195, 5.59644836668897611368267218482, 6.62285458098782667323875065943, 7.982016142944629205708908843182, 8.685082632778970188460998540720, 9.203260032684234440580280948726

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