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Properties

Label 2-1104-1104.827-c0-0-2
Degree $2$
Conductor $1104$
Sign $-0.382 + 0.923i$
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  − i·2-s − 3-s − 4-s + i·6-s + i·8-s + 9-s + 12-s + (1 − i)13-s + 16-s − i·18-s + i·23-s − i·24-s − i·25-s + (−1 − i)26-s − 27-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + i·6-s + i·8-s + 9-s + 12-s + (1 − i)13-s + 16-s − i·18-s + i·23-s − i·24-s − i·25-s + (−1 − i)26-s − 27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6707110475\)
\(L(\frac12)\) \(\approx\) \(0.6707110475\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.998363523545375137200134487878, −9.319320656638941973961056424610, −8.198917838111423497916319565983, −7.44552848209217512570942434110, −5.93412768706794317158196797975, −5.64973236712847979198961112934, −4.37128364227027790406989862501, −3.67473394529503481270957469475, −2.23816043209529402191791397081, −0.815589087383546740846502897119, 1.37967233096274379047021088431, 3.60162475870572270090495352884, 4.50123931895362327892752582808, 5.35884938807010890923412748448, 6.14527007392806418154695545416, 6.89097385186347227428795005523, 7.49345315038614608602989094661, 8.826731530485070996553242006423, 9.147726929406317109049583170881, 10.44747904822497823642558178786

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