Properties

Label 2-1323-21.20-c1-0-25
Degree $2$
Conductor $1323$
Sign $0.755 - 0.654i$
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  + 1.83i·2-s − 1.36·4-s − 3.80·5-s + 1.16i·8-s − 6.98i·10-s − 1.16i·11-s − 5.54i·13-s − 4.86·16-s + 3.17·17-s + 0.631i·19-s + 5.19·20-s + 2.13·22-s − 1.76i·23-s + 9.50·25-s + 10.1·26-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.682·4-s − 1.70·5-s + 0.412i·8-s − 2.20i·10-s − 0.351i·11-s − 1.53i·13-s − 1.21·16-s + 0.770·17-s + 0.144i·19-s + 1.16·20-s + 0.455·22-s − 0.367i·23-s + 1.90·25-s + 1.99·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.024812126\)
\(L(\frac12)\) \(\approx\) \(1.024812126\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.83iT - 2T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
11 \( 1 + 1.16iT - 11T^{2} \)
13 \( 1 + 5.54iT - 13T^{2} \)
17 \( 1 - 3.17T + 17T^{2} \)
19 \( 1 - 0.631iT - 19T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 - 4.83iT - 29T^{2} \)
31 \( 1 + 4.27iT - 31T^{2} \)
37 \( 1 - 4.23T + 37T^{2} \)
41 \( 1 - 4.56T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + 2.54T + 47T^{2} \)
53 \( 1 + 0.0724iT - 53T^{2} \)
59 \( 1 + 6.98T + 59T^{2} \)
61 \( 1 + 8.08iT - 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 - 5.59iT - 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 0.688iT - 97T^{2} \)
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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

−9.410926946753345201984868739879, −8.401812450673517555999448415969, −7.84322991035168180047030148981, −7.58630422203210480327402793900, −6.53881423522370949017263308870, −5.64457648830643036425349376821, −4.84315641366972731604292843634, −3.80169900344041070148985086700, −2.87855309289083544190784370148, −0.55936531763959491094071632986, 0.980921288260781301466566197420, 2.32605997206898775737152561212, 3.45439140550879819626301373808, 4.09369647882330595545681902357, 4.76010202162909877495256207836, 6.39267926055219585898076523317, 7.28051053922899935560748371366, 7.900261772619804341337886571194, 9.011053814124749980191044687375, 9.582595764841833931501094585746

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