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Properties

Label 2-552-552.275-c0-0-4
Degree $2$
Conductor $552$
Sign $-0.5 + 0.866i$
Analytic cond. $0.275483$
Root an. cond. $0.524865$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s − 1.73i·13-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s − 23-s + (0.499 − 0.866i)24-s + 25-s + (−1.49 + 0.866i)26-s − 0.999·27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s − 1.73i·13-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s − 23-s + (0.499 − 0.866i)24-s + 25-s + (−1.49 + 0.866i)26-s − 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(0.275483\)
Root analytic conductor: \(0.524865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7676201396\)
\(L(\frac12)\) \(\approx\) \(0.7676201396\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.60325609197447915695425962406, −9.950548044568939928274613567528, −8.797112719067021130625212655911, −8.198975651525417121704062926254, −7.46956719946119855914312087199, −6.33378024718851112538459611525, −4.94438963742717247654846082095, −3.40375677762047170811127913956, −2.67179232108383802794801508338, −1.16780091408705840809638908167, 2.14562343333243349430060109495, 3.98001731487832688613178207553, 4.69665444278809397578587998070, 5.86572695924130937544020105238, 6.86325955089451224192672683043, 7.87101440579420801326828418201, 8.775365191817477869938895967248, 9.341775679541817493102120213188, 10.13524869501029785898468439814, 10.96642823482114758558468212323

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