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Properties

Label 2-552-184.91-c1-0-33
Degree $2$
Conductor $552$
Sign $-0.265 + 0.963i$
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  + (−1.39 + 0.210i)2-s + 3-s + (1.91 − 0.589i)4-s − 0.707·5-s + (−1.39 + 0.210i)6-s − 4.06·7-s + (−2.54 + 1.22i)8-s + 9-s + (0.989 − 0.148i)10-s + 4.77i·11-s + (1.91 − 0.589i)12-s − 6.61i·13-s + (5.68 − 0.856i)14-s − 0.707·15-s + (3.30 − 2.25i)16-s − 5.50i·17-s + ⋯
L(s)  = 1  + (−0.988 + 0.148i)2-s + 0.577·3-s + (0.955 − 0.294i)4-s − 0.316·5-s + (−0.570 + 0.0860i)6-s − 1.53·7-s + (−0.901 + 0.433i)8-s + 0.333·9-s + (0.312 − 0.0471i)10-s + 1.43i·11-s + (0.551 − 0.170i)12-s − 1.83i·13-s + (1.51 − 0.228i)14-s − 0.182·15-s + (0.826 − 0.563i)16-s − 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.265 + 0.963i$
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: \((2,\ 552,\ (\ :1/2),\ -0.265 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330190 - 0.433623i\)
\(L(\frac12)\) \(\approx\) \(0.330190 - 0.433623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.210i)T \)
3 \( 1 - T \)
23 \( 1 + (-3.15 + 3.61i)T \)
good5 \( 1 + 0.707T + 5T^{2} \)
7 \( 1 + 4.06T + 7T^{2} \)
11 \( 1 - 4.77iT - 11T^{2} \)
13 \( 1 + 6.61iT - 13T^{2} \)
17 \( 1 + 5.50iT - 17T^{2} \)
19 \( 1 + 4.69iT - 19T^{2} \)
29 \( 1 + 1.90iT - 29T^{2} \)
31 \( 1 + 1.05iT - 31T^{2} \)
37 \( 1 + 1.73T + 37T^{2} \)
41 \( 1 + 2.21T + 41T^{2} \)
43 \( 1 + 4.15iT - 43T^{2} \)
47 \( 1 - 4.75iT - 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 3.11T + 59T^{2} \)
61 \( 1 + 5.49T + 61T^{2} \)
67 \( 1 + 9.05iT - 67T^{2} \)
71 \( 1 - 9.05iT - 71T^{2} \)
73 \( 1 - 9.00T + 73T^{2} \)
79 \( 1 - 1.51T + 79T^{2} \)
83 \( 1 - 7.12iT - 83T^{2} \)
89 \( 1 + 7.05iT - 89T^{2} \)
97 \( 1 - 5.21iT - 97T^{2} \)
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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

−10.09075513757746715884533337639, −9.684787261158527458773907753347, −8.932179004973085696759146561645, −7.75175732879390234815296463845, −7.20606150421431641287807393493, −6.32784466832521242647478175960, −4.96255929869845469495923404373, −3.25186125399095877635319908788, −2.52581351510058973120998661742, −0.38530513410163953049756665138, 1.69053813228748093922995427998, 3.27147649517009566573714801560, 3.77518019770929469986056478858, 6.05826821315521602115590671396, 6.57064813123917282299736752202, 7.67449544952807824761321262011, 8.600237311037536737975017533322, 9.226608514611308852496230162590, 9.942231140589881537753238888921, 10.87962471966227314902287687203

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