Properties

Label 2-552-552.275-c0-0-5
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 + (−0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 1.73i·13-s + (−0.5 + 0.866i)16-s − 0.999·18-s + 23-s + (−0.5 + 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 + (−0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 1.73i·13-s + (−0.5 + 0.866i)16-s − 0.999·18-s + 23-s + (−0.5 + 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

\(L(\frac{1}{2})\) \(\approx\) \(1.212292483\)
\(L(\frac12)\) \(\approx\) \(1.212292483\)
\(L(1)\) 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

−11.20116286326183349190252404353, −9.618140212787909615951080502300, −9.215825276884303956023020781896, −8.166045166569601879076225782610, −6.91250590136447026926411748851, −6.21546761745087305879198863546, −4.87177415097243569066680883295, −3.77145034557121457510277612685, −2.59994562385008538924788678468, −1.50922728887907677219645595368, 2.91174480955689051439082182168, 3.63189507226012891018665187190, 5.00272760968178807307386165966, 5.42285308648411887814214878535, 6.80325148517004660573756113774, 7.80668251289516553993309923090, 8.548418488113028763748044728180, 9.301680230692111842299192235517, 10.37890487489081365132608418654, 11.10764694516116745440704018852

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