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Properties

Label 2-552-184.91-c1-0-8
Degree $2$
Conductor $552$
Sign $0.687 - 0.725i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 − 0.723i)2-s + 3-s + (0.952 + 1.75i)4-s − 4.31·5-s + (−1.21 − 0.723i)6-s + 1.64·7-s + (0.114 − 2.82i)8-s + 9-s + (5.23 + 3.11i)10-s + 2.66i·11-s + (0.952 + 1.75i)12-s − 4.33i·13-s + (−2.00 − 1.19i)14-s − 4.31·15-s + (−2.18 + 3.35i)16-s + 0.417i·17-s + ⋯
L(s)  = 1  + (−0.859 − 0.511i)2-s + 0.577·3-s + (0.476 + 0.879i)4-s − 1.92·5-s + (−0.496 − 0.295i)6-s + 0.622·7-s + (0.0406 − 0.999i)8-s + 0.333·9-s + (1.65 + 0.986i)10-s + 0.803i·11-s + (0.275 + 0.507i)12-s − 1.20i·13-s + (−0.534 − 0.318i)14-s − 1.11·15-s + (−0.546 + 0.837i)16-s + 0.101i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.687 - 0.725i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.687 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690692 + 0.297057i\)
\(L(\frac12)\) \(\approx\) \(0.690692 + 0.297057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.723i)T \)
3 \( 1 - T \)
23 \( 1 + (-3.61 - 3.15i)T \)
good5 \( 1 + 4.31T + 5T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 - 2.66iT - 11T^{2} \)
13 \( 1 + 4.33iT - 13T^{2} \)
17 \( 1 - 0.417iT - 17T^{2} \)
19 \( 1 - 7.23iT - 19T^{2} \)
29 \( 1 - 9.62iT - 29T^{2} \)
31 \( 1 - 6.72iT - 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 - 1.83T + 41T^{2} \)
43 \( 1 + 1.68iT - 43T^{2} \)
47 \( 1 - 1.93iT - 47T^{2} \)
53 \( 1 - 5.44T + 53T^{2} \)
59 \( 1 - 9.10T + 59T^{2} \)
61 \( 1 + 0.402T + 61T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 - 1.27iT - 71T^{2} \)
73 \( 1 + 8.90T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 + 18.4iT - 89T^{2} \)
97 \( 1 - 9.76iT - 97T^{2} \)
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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

−10.75343289506918265199740303343, −10.23004533757685943858319574020, −8.825780322034333935683588637110, −8.305846740304114382085723871662, −7.55571621939565766785332472278, −7.09695222380098081379598764865, −4.98431034577283716096122203901, −3.76887744449877441476196058619, −3.15764754231466661490863017129, −1.37381904231531750390969000528, 0.59924994144787402783459454918, 2.56452665265709549143239345698, 4.04493245941247201178130216323, 4.89464362056362025848846451160, 6.60532050371825357024103919526, 7.31241427700734566817126755489, 8.124733374565541982171014381577, 8.647523852900237073392515293256, 9.416802513506525458207846584312, 10.87314058527969178677913553027

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