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Properties

Label 2-552-24.11-c1-0-36
Degree $2$
Conductor $552$
Sign $0.816 + 0.577i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−1 + 1.41i)3-s − 2.00·4-s − 4·5-s + (−2.00 − 1.41i)6-s + 2.82i·7-s − 2.82i·8-s + (−1.00 − 2.82i)9-s − 5.65i·10-s + 5.65i·11-s + (2.00 − 2.82i)12-s − 5.65i·13-s − 4.00·14-s + (4 − 5.65i)15-s + 4.00·16-s + 2.82i·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.577 + 0.816i)3-s − 1.00·4-s − 1.78·5-s + (−0.816 − 0.577i)6-s + 1.06i·7-s − 1.00i·8-s + (−0.333 − 0.942i)9-s − 1.78i·10-s + 1.70i·11-s + (0.577 − 0.816i)12-s − 1.56i·13-s − 1.06·14-s + (1.03 − 1.46i)15-s + 1.00·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1 - 1.41i)T \)
23 \( 1 - T \)
good5 \( 1 + 4T + 5T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5.65iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 2.82iT - 59T^{2} \)
61 \( 1 + 2.82iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + 8.48iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.58078429484708039132182623579, −9.756755912664020849478788683959, −8.700517318933019916393915957414, −8.017511362948952905068003232191, −7.16781318755683762523348533339, −6.02896847189947943260690942753, −5.00827034516637870928338627933, −4.33935575664192211844157380596, −3.31234224582318748799997723500, 0, 1.14087997679507118761902052691, 3.09905732341373504883860162519, 4.05671433118967911994455520199, 4.89890506441801664898339016546, 6.49150672972918407298728380809, 7.40168553661686659987483610786, 8.230920084641081213512856526707, 8.953665078443508240388645990069, 10.53685470125070527016905996099

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