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Properties

Label 2-2243-2243.2242-c0-0-5
Degree $2$
Conductor $2243$
Sign $-1$
Analytic cond. $1.11940$
Root an. cond. $1.05801$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·6-s + 7-s + 1.00·12-s + 1.41i·13-s − 1.41i·14-s − 0.999·16-s − 17-s − 1.41i·19-s − 21-s − 1.41i·23-s + 25-s + 2.00·26-s + 27-s + ⋯
L(s)  = 1  − 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·6-s + 7-s + 1.00·12-s + 1.41i·13-s − 1.41i·14-s − 0.999·16-s − 17-s − 1.41i·19-s − 21-s − 1.41i·23-s + 25-s + 2.00·26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2243\)
Sign: $-1$
Analytic conductor: \(1.11940\)
Root analytic conductor: \(1.05801\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2243} (2242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2243,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7099385350\)
\(L(\frac12)\) \(\approx\) \(0.7099385350\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2243 \( 1+O(T) \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.962393933796387657365311424546, −8.482199922414724606848310019976, −6.94352596521534409344545396053, −6.65827870808607036557867147840, −5.37567691996034705903213635450, −4.55728277366969868345862911955, −4.12953025014951936411201534367, −2.58769562199968904272912003106, −1.96616399236372515750404622028, −0.56997466504479186504109130291, 1.49715501192150369592956069868, 3.12727354174716180371909470310, 4.51918019933788515608940550875, 5.33740031722516181176432180321, 5.54160177370721695696376797694, 6.45226372269689607501465298815, 7.18820229746084786660285986018, 8.028405585152080278133184181256, 8.398170831540644111256668012385, 9.357508518898910300656862896331

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