Properties

Label 2-552-1.1-c1-0-1
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 − 4.42·5-s + 0.622·7-s + 9-s + 5.80·11-s + 2·13-s − 4.42·15-s + 1.37·17-s + 6.42·19-s + 0.622·21-s − 23-s + 14.6·25-s + 27-s − 0.755·29-s − 1.24·31-s + 5.80·33-s − 2.75·35-s + 9.05·37-s + 2·39-s − 9.61·41-s − 5.18·43-s − 4.42·45-s − 5.24·47-s − 6.61·49-s + 1.37·51-s + 12.0·53-s − 25.7·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.98·5-s + 0.235·7-s + 0.333·9-s + 1.75·11-s + 0.554·13-s − 1.14·15-s + 0.334·17-s + 1.47·19-s + 0.135·21-s − 0.208·23-s + 2.92·25-s + 0.192·27-s − 0.140·29-s − 0.223·31-s + 1.01·33-s − 0.465·35-s + 1.48·37-s + 0.320·39-s − 1.50·41-s − 0.790·43-s − 0.660·45-s − 0.764·47-s − 0.944·49-s + 0.192·51-s + 1.65·53-s − 3.46·55-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

\(L(1)\) \(\approx\) \(1.473761101\)
\(L(\frac12)\) \(\approx\) \(1.473761101\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.42T + 5T^{2} \)
7 \( 1 - 0.622T + 7T^{2} \)
11 \( 1 - 5.80T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 - 6.42T + 19T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 - 9.05T + 37T^{2} \)
41 \( 1 + 9.61T + 41T^{2} \)
43 \( 1 + 5.18T + 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 2.94T + 61T^{2} \)
67 \( 1 - 5.18T + 67T^{2} \)
71 \( 1 - 6.10T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 8.62T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 6.23T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
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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

−11.09350062267688387102878173990, −9.741158672248007240619437629905, −8.835206714784139475157934830658, −8.108130592088647416839368747514, −7.38744311685683997301004653824, −6.51881503811754765688253498091, −4.84425447278574149094917013672, −3.81422658709136073325009122003, −3.34314264910790591298518562710, −1.16847977518486291489816016315, 1.16847977518486291489816016315, 3.34314264910790591298518562710, 3.81422658709136073325009122003, 4.84425447278574149094917013672, 6.51881503811754765688253498091, 7.38744311685683997301004653824, 8.108130592088647416839368747514, 8.835206714784139475157934830658, 9.741158672248007240619437629905, 11.09350062267688387102878173990

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