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Properties

Label 2-1323-63.20-c1-0-28
Degree $2$
Conductor $1323$
Sign $-0.790 - 0.612i$
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  + (−1.58 − 0.916i)2-s + (0.678 + 1.17i)4-s + (−0.322 − 0.559i)5-s + 1.17i·8-s + 1.18i·10-s + (−4.60 − 2.65i)11-s + (4.44 − 2.56i)13-s + (2.43 − 4.22i)16-s − 1.62·17-s + 2.41i·19-s + (0.437 − 0.758i)20-s + (4.86 + 8.43i)22-s + (1.27 − 0.735i)23-s + (2.29 − 3.96i)25-s − 9.39·26-s + ⋯
L(s)  = 1  + (−1.12 − 0.647i)2-s + (0.339 + 0.587i)4-s + (−0.144 − 0.250i)5-s + 0.416i·8-s + 0.374i·10-s + (−1.38 − 0.801i)11-s + (1.23 − 0.711i)13-s + (0.609 − 1.05i)16-s − 0.395·17-s + 0.553i·19-s + (0.0978 − 0.169i)20-s + (1.03 + 1.79i)22-s + (0.265 − 0.153i)23-s + (0.458 − 0.793i)25-s − 1.84·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2267335898\)
\(L(\frac12)\) \(\approx\) \(0.2267335898\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.916i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.322 + 0.559i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.60 + 2.65i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.44 + 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 - 2.41iT - 19T^{2} \)
23 \( 1 + (-1.27 + 0.735i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.43 - 3.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.90 - 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + (5.99 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.54 + 2.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.36iT - 53T^{2} \)
59 \( 1 + (1.47 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.18 + 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.51 - 6.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (14.3 + 8.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.950028829236967508753816651790, −8.462395513796266736934568508135, −8.047924707369815419398838621715, −6.88205652671995943790488253596, −5.67025994822465748702552853756, −5.05496321799715032549286768781, −3.52885088318049426178655915129, −2.66694522831071891261278196272, −1.35009210409037648713122612400, −0.14721454826645136995946740515, 1.55261340962847305667038430301, 2.98157477928190568028430134782, 4.18272277526212312643412211394, 5.23491770016338813558171542467, 6.40785210743039986686702599847, 7.00196881960831895433657377933, 7.75881987878522281285122636169, 8.486901990780288993109769574448, 9.146185532135903157672421892316, 9.955551600608406296255071049995

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