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Properties

Label 2-2112-264.131-c0-0-4
Degree $2$
Conductor $2112$
Sign $0.707 + 0.707i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + i·3-s − 2·5-s − 9-s − i·11-s − 2i·15-s + 3·25-s − i·27-s − 2i·31-s + 33-s + 2·45-s − 49-s + 2·53-s + 2i·55-s − 2i·59-s + 3i·75-s + ⋯
L(s)  = 1  + i·3-s − 2·5-s − 9-s − i·11-s − 2i·15-s + 3·25-s − i·27-s − 2i·31-s + 33-s + 2·45-s − 49-s + 2·53-s + 2i·55-s − 2i·59-s + 3i·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5259400753\)
\(L(\frac12)\) \(\approx\) \(0.5259400753\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
11 \( 1 + iT \)
good5 \( 1 + 2T + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−9.073895362388300972684184461219, −8.297462521291868234662333139723, −7.964392068154133031584569531419, −6.96571197397685603904722538685, −5.91604422686958576617899832468, −4.96273199366801337508575170129, −4.10840789586996804490278873470, −3.63684706714654218528068858797, −2.78304019172745381435391708830, −0.42619479697259562785742724431, 1.20502155747553865866092001941, 2.66141211220329146021529683439, 3.57767006366385665176881307268, 4.47368810841854421858031062957, 5.32152270710470037201934449704, 6.69256370088733173970323250153, 7.12966254227466381312449137927, 7.72592419917071722329831796338, 8.437011734808657398599220184313, 9.000517353135505745239014386618

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