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Properties

Label 2-1104-1104.275-c0-0-2
Degree $2$
Conductor $1104$
Sign $0.923 - 0.382i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (0.366 + 0.366i)13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + i·23-s + (0.499 − 0.866i)24-s + i·25-s + (−0.5 − 0.133i)26-s − 0.999i·27-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.499i)12-s + (0.366 + 0.366i)13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + i·23-s + (0.499 − 0.866i)24-s + i·25-s + (−0.5 − 0.133i)26-s − 0.999i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5339338826\)
\(L(\frac12)\) \(\approx\) \(0.5339338826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - iT \)
good5 \( 1 - iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.73T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
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

−9.989277141020695825916440958493, −9.336155704489500293183347913577, −8.328439170696268085289094074177, −7.48798350038660212789109323485, −6.92708072689283233344423774153, −5.91886927251732852257925010989, −5.44400593633297932527337132508, −4.17242833116130826884139134974, −2.33980725656196735187027177191, −1.12099097092326100186523868001, 0.928578490718459885996224900777, 2.57699348925854865483778556843, 3.77565763228613140493452055235, 4.66942774344026974642193237903, 5.91778301761860912291947015656, 6.66773799469694462218209029996, 7.59323031575867738679984832486, 8.637284979500861446228282475900, 9.222396602656160078327937145137, 10.37194769638986752996856962616

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