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Properties

Label 2-552-8.5-c1-0-21
Degree $2$
Conductor $552$
Sign $0.990 + 0.134i$
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  + (−1.19 − 0.761i)2-s + i·3-s + (0.839 + 1.81i)4-s + 1.20i·5-s + (0.761 − 1.19i)6-s + 3.87·7-s + (0.381 − 2.80i)8-s − 9-s + (0.917 − 1.43i)10-s − 6.13i·11-s + (−1.81 + 0.839i)12-s − 3.14i·13-s + (−4.61 − 2.95i)14-s − 1.20·15-s + (−2.58 + 3.04i)16-s + 3.86·17-s + ⋯
L(s)  = 1  + (−0.842 − 0.538i)2-s + 0.577i·3-s + (0.419 + 0.907i)4-s + 0.538i·5-s + (0.310 − 0.486i)6-s + 1.46·7-s + (0.134 − 0.990i)8-s − 0.333·9-s + (0.290 − 0.453i)10-s − 1.84i·11-s + (−0.523 + 0.242i)12-s − 0.872i·13-s + (−1.23 − 0.788i)14-s − 0.310·15-s + (−0.647 + 0.762i)16-s + 0.937·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.990 + 0.134i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16322 - 0.0788414i\)
\(L(\frac12)\) \(\approx\) \(1.16322 - 0.0788414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.761i)T \)
3 \( 1 - iT \)
23 \( 1 - T \)
good5 \( 1 - 1.20iT - 5T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 + 6.13iT - 11T^{2} \)
13 \( 1 + 3.14iT - 13T^{2} \)
17 \( 1 - 3.86T + 17T^{2} \)
19 \( 1 - 4.21iT - 19T^{2} \)
29 \( 1 - 1.40iT - 29T^{2} \)
31 \( 1 - 4.44T + 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + 8.51T + 41T^{2} \)
43 \( 1 - 8.39iT - 43T^{2} \)
47 \( 1 + 0.0559T + 47T^{2} \)
53 \( 1 + 0.837iT - 53T^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 + 8.37iT - 61T^{2} \)
67 \( 1 - 9.25iT - 67T^{2} \)
71 \( 1 - 4.54T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 9.67T + 79T^{2} \)
83 \( 1 + 6.17iT - 83T^{2} \)
89 \( 1 - 9.45T + 89T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−10.64642290595487577262468738242, −10.22707828778685882069877351367, −8.902022577228042188129235651422, −8.216983776260689601316415608795, −7.68478662618759898745872768466, −6.18225244666657179859036738805, −5.14204941669701363054812402157, −3.66732419576262244385886746002, −2.87072180808143548532752645929, −1.14427748662975375678260086910, 1.30283441656697985486020483369, 2.16655403730775269098524703403, 4.69913945644007586228171318706, 5.10564993689111937837902605388, 6.64259623076412938982235116703, 7.31603337575991178522248486927, 8.098318307638509820268901004031, 8.865269048546702147803050808900, 9.742605509242728260663812121018, 10.68828184031468037582538788594

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