Properties

Label 2-1323-63.20-c1-0-16
Degree $2$
Conductor $1323$
Sign $-0.816 + 0.577i$
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  + (−2.23 − 1.28i)2-s + (2.32 + 4.02i)4-s + (−1.16 − 2.01i)5-s − 6.82i·8-s + 6.01i·10-s + (3.78 + 2.18i)11-s + (−1.14 + 0.660i)13-s + (−4.15 + 7.18i)16-s − 5.78·17-s + 0.675i·19-s + (5.41 − 9.38i)20-s + (−5.62 − 9.74i)22-s + (4.81 − 2.78i)23-s + (−0.218 + 0.379i)25-s + 3.40·26-s + ⋯
L(s)  = 1  + (−1.57 − 0.911i)2-s + (1.16 + 2.01i)4-s + (−0.521 − 0.903i)5-s − 2.41i·8-s + 1.90i·10-s + (1.14 + 0.658i)11-s + (−0.317 + 0.183i)13-s + (−1.03 + 1.79i)16-s − 1.40·17-s + 0.154i·19-s + (1.21 − 2.09i)20-s + (−1.19 − 2.07i)22-s + (1.00 − 0.580i)23-s + (−0.0437 + 0.0758i)25-s + 0.667·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.816 + 0.577i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5244413507\)
\(L(\frac12)\) \(\approx\) \(0.5244413507\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 1.28i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.78 - 2.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.14 - 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 - 0.675iT - 19T^{2} \)
23 \( 1 + (-4.81 + 2.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.86 + 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.47 + 2.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + (-3.29 - 5.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.89 + 6.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.246 - 0.427i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.14iT - 53T^{2} \)
59 \( 1 + (2.15 + 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.77 - 1.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.32 + 9.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.32T + 89T^{2} \)
97 \( 1 + (12.7 + 7.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.140842462796378452981783265841, −8.882175582804976250707538863492, −7.993071769164859968848209236384, −7.20655904690347963478756225985, −6.40032891311720249882842884827, −4.65454639698981507615498273335, −4.00971544782854197274837713878, −2.62883814085162643535312993994, −1.60172594573548004686061348392, −0.44963755344226189185047714538, 1.08595064753233438825264125580, 2.56123354072555847352485926458, 3.86091283606483411092336386091, 5.29199142966289045296368686008, 6.40648619446813024000988085981, 6.82234851265331932248300217134, 7.49592563159043959874059480249, 8.361550092605538425096560066253, 9.123291110042119755697779275391, 9.569189593628404241322994163920

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