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Properties

Label 2-1323-9.4-c1-0-33
Degree $2$
Conductor $1323$
Sign $-0.999 + 0.0167i$
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  + (−0.119 − 0.207i)2-s + (0.971 − 1.68i)4-s + (−0.590 + 1.02i)5-s − 0.942·8-s + 0.282·10-s + (−1.85 − 3.20i)11-s + (0.5 − 0.866i)13-s + (−1.83 − 3.16i)16-s − 6.94·17-s − 1.94·19-s + (1.14 + 1.98i)20-s + (−0.442 + 0.766i)22-s + (−2.80 + 4.85i)23-s + (1.80 + 3.12i)25-s − 0.239·26-s + ⋯
L(s)  = 1  + (−0.0845 − 0.146i)2-s + (0.485 − 0.841i)4-s + (−0.264 + 0.457i)5-s − 0.333·8-s + 0.0893·10-s + (−0.558 − 0.967i)11-s + (0.138 − 0.240i)13-s + (−0.457 − 0.792i)16-s − 1.68·17-s − 0.445·19-s + (0.256 + 0.444i)20-s + (−0.0944 + 0.163i)22-s + (−0.584 + 1.01i)23-s + (0.360 + 0.624i)25-s − 0.0468·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.0167i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.999 + 0.0167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5159785577\)
\(L(\frac12)\) \(\approx\) \(0.5159785577\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.119 + 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.590 - 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.85 + 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
23 \( 1 + (2.80 - 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.830 + 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (-5.09 + 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.80 + 6.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 - 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 + (-3.58 - 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.161299804471134239421360270473, −8.602450444737164835337061699275, −7.44016361074956251343239558189, −6.76080188700820285185476559340, −5.88866793285405164900590162555, −5.20449903360357476104239078869, −3.90687193712974417022186936887, −2.84376931034338381081309340567, −1.83683858060428723570752233424, −0.19631850582741353447471422301, 1.97578825152416051364773576832, 2.83845913256018859175305564517, 4.28069310422287707472236740358, 4.62161097739201357511159715691, 6.15011558270532929779630321003, 6.84594368225253421008626574079, 7.58592027754514513069646533788, 8.546726761229560217540565075980, 8.830843033718568675117327120351, 10.09301192268495585925020794579

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