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Properties

Label 2-1323-9.4-c1-0-3
Degree $2$
Conductor $1323$
Sign $-0.544 + 0.839i$
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  + (1.08 + 1.88i)2-s + (−1.36 + 2.36i)4-s + (−0.634 + 1.09i)5-s − 1.60·8-s − 2.76·10-s + (−2.73 − 4.74i)11-s + (−2.37 + 4.10i)13-s + (0.992 + 1.71i)16-s − 4.81·17-s − 5.38·19-s + (−1.73 − 3.00i)20-s + (5.96 − 10.3i)22-s + (−2.58 + 4.48i)23-s + (1.69 + 2.93i)25-s − 10.3·26-s + ⋯
L(s)  = 1  + (0.769 + 1.33i)2-s + (−0.684 + 1.18i)4-s + (−0.283 + 0.491i)5-s − 0.566·8-s − 0.872·10-s + (−0.825 − 1.43i)11-s + (−0.658 + 1.13i)13-s + (0.248 + 0.429i)16-s − 1.16·17-s − 1.23·19-s + (−0.388 − 0.672i)20-s + (1.27 − 2.20i)22-s + (−0.539 + 0.934i)23-s + (0.339 + 0.587i)25-s − 2.02·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.544 + 0.839i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.544 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.011766872\)
\(L(\frac12)\) \(\approx\) \(1.011766872\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 - 1.88i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.634 - 1.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 + 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.37 - 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.81T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + (2.58 - 4.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.732 - 1.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.91T + 37T^{2} \)
41 \( 1 + (1.94 - 3.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.66 + 2.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.57 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + (0.154 - 0.267i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.17 - 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.08 + 8.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (-2.48 - 4.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.18626634000123615934007984259, −8.944499435856710758122231975098, −8.343106824538837806579648900659, −7.46908496250315035702537823886, −6.79627007674271489241541035575, −6.11128119573303278376281062011, −5.29424267613616576171771656157, −4.35790122775283400694211706387, −3.56719633596911434734270843354, −2.24650532364308995836275859714, 0.29452876400649993470076675388, 2.05894660019676972319187638783, 2.58601698154981422259884632438, 3.92941251624425750439734731540, 4.70485708030749657229899151908, 5.13179123134558156087008872725, 6.49883069247848566908545369015, 7.57564798822325246410269210923, 8.351113035229380250582253746872, 9.396835109921674099288137482349

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