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Properties

Label 2-1107-123.122-c0-0-4
Degree $2$
Conductor $1107$
Sign $-1$
Analytic cond. $0.552464$
Root an. cond. $0.743279$
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 − 1.00·4-s − 1.41i·5-s + 1.41i·7-s − 2.00·10-s − 11-s − 1.41i·13-s + 2.00·14-s − 0.999·16-s − 1.41i·19-s + 1.41i·20-s + 1.41i·22-s − 1.00·25-s − 2.00·26-s − 1.41i·28-s − 29-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 1.41i·5-s + 1.41i·7-s − 2.00·10-s − 11-s − 1.41i·13-s + 2.00·14-s − 0.999·16-s − 1.41i·19-s + 1.41i·20-s + 1.41i·22-s − 1.00·25-s − 2.00·26-s − 1.41i·28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1107\)    =    \(3^{3} \cdot 41\)
Sign: $-1$
Analytic conductor: \(0.552464\)
Root analytic conductor: \(0.743279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1107} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1107,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8925931439\)
\(L(\frac12)\) \(\approx\) \(0.8925931439\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.693355868793239995780073759769, −8.880960795876378647596155501130, −8.525907394828604121291160437184, −7.38132001661382150439374460797, −5.71886491624443781204562777416, −5.25067536313413623600156374438, −4.31052167027252218128965306450, −2.90516079165178273107100149442, −2.31852257281775222878454378633, −0.807456907444710392592549952266, 2.21101525323922509061381103253, 3.63699668200224752232938513318, 4.47520831862296695585442930635, 5.73074845961126810791526048902, 6.46579181124376247314890587282, 7.32980150248204513539124427537, 7.42369120338581171711527888145, 8.453667121427858262698861795308, 9.643000994280934489844002966681, 10.46663274160680697982071018792

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