Properties

Label 2-1323-63.4-c1-0-0
Degree $2$
Conductor $1323$
Sign $-0.515 + 0.856i$
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  + (−0.551 + 0.955i)2-s + (0.391 + 0.678i)4-s − 0.105·5-s − 3.07·8-s + (0.0581 − 0.100i)10-s − 3.33·11-s + (−1.23 + 2.14i)13-s + (0.909 − 1.57i)16-s + (0.806 − 1.39i)17-s + (3.84 + 6.65i)19-s + (−0.0413 − 0.0715i)20-s + (1.84 − 3.18i)22-s + 1.89·23-s − 4.98·25-s + (−1.36 − 2.36i)26-s + ⋯
L(s)  = 1  + (−0.389 + 0.675i)2-s + (0.195 + 0.339i)4-s − 0.0471·5-s − 1.08·8-s + (0.0183 − 0.0318i)10-s − 1.00·11-s + (−0.343 + 0.595i)13-s + (0.227 − 0.393i)16-s + (0.195 − 0.338i)17-s + (0.881 + 1.52i)19-s + (−0.00924 − 0.0160i)20-s + (0.392 − 0.679i)22-s + 0.395·23-s − 0.997·25-s + (−0.268 − 0.464i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.515 + 0.856i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2169469162\)
\(L(\frac12)\) \(\approx\) \(0.2169469162\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.551 - 0.955i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.105T + 5T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.806 + 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.84 - 6.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.63 + 8.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.59 - 2.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.98 - 8.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.22 - 3.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.584 - 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.695962244983025703422797340946, −9.533166019645848971655621464247, −8.155088867540074957261911654691, −7.82437124476764411569209317072, −7.14970326689605125692769906792, −6.03817278273476803894839552014, −5.48501790445009436441059429845, −4.14877612853405647654250909858, −3.15077531757250844360015611363, −2.03020935017676620851211511502, 0.095412219539733853448087797544, 1.54228321320418329936668592982, 2.71720449862479353619081792612, 3.44302416439731652614092546794, 5.16346608033450223952223059221, 5.42659043847161901039087381259, 6.77256155852632955707808819680, 7.44308773693445104881724109925, 8.507841197789271820852411601301, 9.254458883751526717609222155225

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