Properties

Label 2-1323-21.20-c1-0-31
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  + 0.656i·2-s + 1.56·4-s + 3.31·5-s + 2.34i·8-s + 2.17i·10-s − 2.34i·11-s + 1.58i·13-s + 1.59·16-s + 1.13·17-s − 4.44i·19-s + 5.19·20-s + 1.53·22-s + 9.39i·23-s + 5.96·25-s − 1.03·26-s + ⋯
L(s)  = 1  + 0.464i·2-s + 0.784·4-s + 1.48·5-s + 0.828i·8-s + 0.687i·10-s − 0.706i·11-s + 0.438i·13-s + 0.399·16-s + 0.275·17-s − 1.02i·19-s + 1.16·20-s + 0.328·22-s + 1.95i·23-s + 1.19·25-s − 0.203·26-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

\(L(1)\) \(\approx\) \(2.737333788\)
\(L(\frac12)\) \(\approx\) \(2.737333788\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.656iT - 2T^{2} \)
5 \( 1 - 3.31T + 5T^{2} \)
11 \( 1 + 2.34iT - 11T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 - 1.13T + 17T^{2} \)
19 \( 1 + 4.44iT - 19T^{2} \)
23 \( 1 - 9.39iT - 23T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + 7.31iT - 31T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 - 9.64T + 41T^{2} \)
43 \( 1 + 2.16T + 43T^{2} \)
47 \( 1 + 5.58T + 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 + 4.00iT - 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 6.39iT - 71T^{2} \)
73 \( 1 + 10.6iT - 73T^{2} \)
79 \( 1 - 1.23T + 79T^{2} \)
83 \( 1 + 1.03T + 83T^{2} \)
89 \( 1 - 7.47T + 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−9.576457071938940324277610300600, −9.063110461659109247962790824035, −7.975581051692301811328029981081, −7.12629595080731366097745989707, −6.34944525587853487003650751921, −5.70317951900057604290552758170, −5.07559520594492684912470380527, −3.43301754348880999556572626976, −2.39140283669171697157005521038, −1.48178921500680867280685967511, 1.32996791150708728023108016905, 2.22312869077430032336123706928, 3.01371979984636755571761968739, 4.37023477019508173335943946636, 5.53314344682375119632686977674, 6.21904360053033021775130461403, 6.90132237097931043554206503427, 7.892289405685730305549323116854, 8.931522114478221334000436651063, 9.858161051744876984708242462520

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