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Properties

Label 2-1104-1104.827-c0-0-3
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  + i·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s + i·12-s + (1 − i)13-s + 16-s − i·18-s − i·23-s − 24-s − i·25-s + (1 + i)26-s + i·27-s + ⋯
L(s)  = 1  + i·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s + i·12-s + (1 − i)13-s + 16-s − i·18-s − i·23-s − 24-s − i·25-s + (1 + i)26-s + i·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: $\chi_{1104} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8963594521\)
\(L(\frac12)\) \(\approx\) \(0.8963594521\)
\(L(1)\) 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 \( 1 + iT \)
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

−9.873738888630128494489335998468, −8.625888226060976460508526230496, −8.339219614178581690611389782229, −7.51207712404278987016517739194, −6.56820439090432198566923588825, −6.05148595446201581980349313352, −5.15965159723940255656643051507, −3.94191660277821711646236674385, −2.69631262354301774915516536310, −0.915504670663884908587965140384, 1.62722133825104244685436381857, 3.08760447423791260672461362255, 3.78559143736109042547736897636, 4.69730497101381016813393064313, 5.49642998074701204040540971248, 6.59592396407637288282397194931, 8.059119333395022455122198764954, 8.809020359058085170396494193831, 9.406433429271158619751818383608, 10.14557909251180529172325303703

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