Properties

Label 2-552-1.1-c1-0-4
Degree $2$
Conductor $552$
Sign $1$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.35·5-s − 2.96·7-s + 9-s + 1.61·11-s + 2·13-s + 3.35·15-s + 4.96·17-s − 1.35·19-s − 2.96·21-s − 23-s + 6.22·25-s + 27-s − 7.92·29-s + 5.92·31-s + 1.61·33-s − 9.92·35-s − 2.31·37-s + 2·39-s − 1.22·41-s − 4.57·43-s + 3.35·45-s + 1.92·47-s + 1.77·49-s + 4.96·51-s − 4.12·53-s + 5.40·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.49·5-s − 1.11·7-s + 0.333·9-s + 0.486·11-s + 0.554·13-s + 0.865·15-s + 1.20·17-s − 0.309·19-s − 0.646·21-s − 0.208·23-s + 1.24·25-s + 0.192·27-s − 1.47·29-s + 1.06·31-s + 0.280·33-s − 1.67·35-s − 0.380·37-s + 0.320·39-s − 0.191·41-s − 0.697·43-s + 0.499·45-s + 0.280·47-s + 0.253·49-s + 0.694·51-s − 0.566·53-s + 0.728·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.108954809\)
\(L(\frac12)\) \(\approx\) \(2.108954809\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 - 3.35T + 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 + 4.57T + 43T^{2} \)
47 \( 1 - 1.92T + 47T^{2} \)
53 \( 1 + 4.12T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 + 16.6T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 5.03T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.36575622704556358273351661243, −9.823837300450310182802435924752, −9.251210399008956117268990116946, −8.314857791562294143984530421266, −7.00304345149988642058209307039, −6.20019300546419628763333340518, −5.45420201330113245774286302838, −3.83865868387026356686091511439, −2.83154157020359856438247917147, −1.54850387854144576745246497437, 1.54850387854144576745246497437, 2.83154157020359856438247917147, 3.83865868387026356686091511439, 5.45420201330113245774286302838, 6.20019300546419628763333340518, 7.00304345149988642058209307039, 8.314857791562294143984530421266, 9.251210399008956117268990116946, 9.823837300450310182802435924752, 10.36575622704556358273351661243

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