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Properties

Label 2-1323-21.20-c1-0-22
Degree $2$
Conductor $1323$
Sign $0.654 - 0.755i$
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  + 2·4-s + 6.92i·13-s + 4·16-s + 5.19i·19-s − 5·25-s + 1.73i·31-s + 10·37-s + 13·43-s + 13.8i·52-s − 8.66i·61-s + 8·64-s − 16·67-s − 15.5i·73-s + 10.3i·76-s − 4·79-s + ⋯
L(s)  = 1  + 4-s + 1.92i·13-s + 16-s + 1.19i·19-s − 25-s + 0.311i·31-s + 1.64·37-s + 1.98·43-s + 1.92i·52-s − 1.10i·61-s + 64-s − 1.95·67-s − 1.82i·73-s + 1.19i·76-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.066968595\)
\(L(\frac12)\) \(\approx\) \(2.066968595\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−9.720545078660926337409100086332, −9.047882339638663826006534434304, −7.901546385957358706369035545449, −7.34390439351496400923478523582, −6.34212094823314777583954998237, −5.91000046594208720490185466022, −4.51653511898605822446560390381, −3.66231900056800214116411185518, −2.38070878805577333716580364620, −1.53280133028891762517820384060, 0.870944079429150810642027048923, 2.43508036056688723772277769535, 3.09137629284285801829700379557, 4.35148728407102956312501317412, 5.64130492117720509434547426052, 6.03300846792577051191613937301, 7.28000444148362533266653753107, 7.68359056743721863298099230779, 8.596756457728321181846522949344, 9.689728419362521521300834171622

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