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Properties

Label 2-1323-63.20-c1-0-29
Degree $2$
Conductor $1323$
Sign $-0.999 + 0.00155i$
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  + (−0.621 − 0.359i)2-s + (−0.742 − 1.28i)4-s + (0.723 + 1.25i)5-s + 2.50i·8-s − 1.03i·10-s + (−1.55 − 0.900i)11-s + (−1.88 + 1.09i)13-s + (−0.585 + 1.01i)16-s + 3.90·17-s − 4.01i·19-s + (1.07 − 1.86i)20-s + (0.646 + 1.11i)22-s + (−4.91 + 2.83i)23-s + (1.45 − 2.51i)25-s + 1.56·26-s + ⋯
L(s)  = 1  + (−0.439 − 0.253i)2-s + (−0.371 − 0.642i)4-s + (0.323 + 0.560i)5-s + 0.884i·8-s − 0.328i·10-s + (−0.470 − 0.271i)11-s + (−0.523 + 0.302i)13-s + (−0.146 + 0.253i)16-s + 0.947·17-s − 0.920i·19-s + (0.240 − 0.416i)20-s + (0.137 + 0.238i)22-s + (−1.02 + 0.591i)23-s + (0.290 − 0.503i)25-s + 0.307·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.00155i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.999 + 0.00155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2645030121\)
\(L(\frac12)\) \(\approx\) \(0.2645030121\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.621 + 0.359i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.723 - 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.55 + 0.900i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.88 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 + 4.01iT - 19T^{2} \)
23 \( 1 + (4.91 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.823T + 37T^{2} \)
41 \( 1 + (5.90 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 + 2.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.15iT - 53T^{2} \)
59 \( 1 + (4.89 + 8.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.03 - 1.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.156 - 0.270i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + 2.80iT - 73T^{2} \)
79 \( 1 + (6.21 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.60 - 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (13.4 + 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.544990258212795654862327428448, −8.519721177073814526644146394626, −7.71320490131789544027415188726, −6.77641677715062676740105875978, −5.74894380317841492428109244235, −5.20638280614142331991325857762, −3.99879372149732033535013257760, −2.71168509696724299372864508610, −1.74828047137906055420398678138, −0.12527237557891822934794697130, 1.58022432492689337069528157752, 3.06429153059450682307614014419, 4.01199260948185589651148512541, 5.06640742306209990035956210270, 5.80387732233368571875779518098, 7.04549489775618282320650382976, 7.75810331877842531416403731741, 8.329762427710302391560808253453, 9.222797834436964563360352363549, 9.849570366119354930218688451558

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