Properties

Label 2-552-1.1-c1-0-5
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 + 1.07·5-s + 4.34·7-s + 9-s − 3.41·11-s + 2·13-s + 1.07·15-s − 2.34·17-s + 0.921·19-s + 4.34·21-s − 23-s − 3.83·25-s + 27-s + 6.68·29-s − 8.68·31-s − 3.41·33-s + 4.68·35-s + 7.26·37-s + 2·39-s + 8.83·41-s + 7.75·43-s + 1.07·45-s − 12.6·47-s + 11.8·49-s − 2.34·51-s − 11.9·53-s − 3.68·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.482·5-s + 1.64·7-s + 0.333·9-s − 1.03·11-s + 0.554·13-s + 0.278·15-s − 0.567·17-s + 0.211·19-s + 0.947·21-s − 0.208·23-s − 0.767·25-s + 0.192·27-s + 1.24·29-s − 1.55·31-s − 0.595·33-s + 0.791·35-s + 1.19·37-s + 0.320·39-s + 1.38·41-s + 1.18·43-s + 0.160·45-s − 1.84·47-s + 1.69·49-s − 0.327·51-s − 1.63·53-s − 0.497·55-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

\(L(1)\) \(\approx\) \(2.131650832\)
\(L(\frac12)\) \(\approx\) \(2.131650832\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.07T + 5T^{2} \)
7 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 - 0.921T + 19T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 - 7.26T + 37T^{2} \)
41 \( 1 - 8.83T + 41T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 1.84T + 59T^{2} \)
61 \( 1 - 4.73T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 - 2.52T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 0.894T + 83T^{2} \)
89 \( 1 + 8.49T + 89T^{2} \)
97 \( 1 - 8.15T + 97T^{2} \)
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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

−10.90943717741837721418520657492, −9.879093100675415401235072526782, −8.917382188033609842943386201798, −8.049204183291778142387603396240, −7.55969747885931562150524405727, −6.10774384929442049482515829042, −5.10473166142377301052699449770, −4.20327172086876473975540025475, −2.64136067879323748687121798124, −1.59750872432190142870419687390, 1.59750872432190142870419687390, 2.64136067879323748687121798124, 4.20327172086876473975540025475, 5.10473166142377301052699449770, 6.10774384929442049482515829042, 7.55969747885931562150524405727, 8.049204183291778142387603396240, 8.917382188033609842943386201798, 9.879093100675415401235072526782, 10.90943717741837721418520657492

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