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Properties

Label 2-552-8.5-c1-0-13
Degree $2$
Conductor $552$
Sign $0.959 + 0.280i$
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  + (0.133 − 1.40i)2-s − i·3-s + (−1.96 − 0.376i)4-s + 2.81i·5-s + (−1.40 − 0.133i)6-s + 1.52·7-s + (−0.793 + 2.71i)8-s − 9-s + (3.96 + 0.377i)10-s + 5.55i·11-s + (−0.376 + 1.96i)12-s − 1.85i·13-s + (0.204 − 2.15i)14-s + 2.81·15-s + (3.71 + 1.48i)16-s + 5.32·17-s + ⋯
L(s)  = 1  + (0.0946 − 0.995i)2-s − 0.577i·3-s + (−0.982 − 0.188i)4-s + 1.26i·5-s + (−0.574 − 0.0546i)6-s + 0.578·7-s + (−0.280 + 0.959i)8-s − 0.333·9-s + (1.25 + 0.119i)10-s + 1.67i·11-s + (−0.108 + 0.567i)12-s − 0.514i·13-s + (0.0547 − 0.575i)14-s + 0.727·15-s + (0.929 + 0.370i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.959 + 0.280i$
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: \((2,\ 552,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35802 - 0.194334i\)
\(L(\frac12)\) \(\approx\) \(1.35802 - 0.194334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.133 + 1.40i)T \)
3 \( 1 + iT \)
23 \( 1 - T \)
good5 \( 1 - 2.81iT - 5T^{2} \)
7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 - 5.55iT - 11T^{2} \)
13 \( 1 + 1.85iT - 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 3.73iT - 19T^{2} \)
29 \( 1 - 0.0915iT - 29T^{2} \)
31 \( 1 + 5.72T + 31T^{2} \)
37 \( 1 + 4.54iT - 37T^{2} \)
41 \( 1 - 6.21T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 - 5.98T + 47T^{2} \)
53 \( 1 + 6.56iT - 53T^{2} \)
59 \( 1 - 3.88iT - 59T^{2} \)
61 \( 1 - 7.32iT - 61T^{2} \)
67 \( 1 + 15.8iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 0.668T + 73T^{2} \)
79 \( 1 - 8.30T + 79T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 - 6.22T + 89T^{2} \)
97 \( 1 - 8.93T + 97T^{2} \)
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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

−10.72289688600822758929040429654, −10.15321297672199229137930513265, −9.283733782189270263975865211463, −7.83017656286928541780021938417, −7.44546514387299206623109716059, −6.09090928203334780541660184951, −5.02035112709787313693092392608, −3.72611981683768521670166926422, −2.63806844467327113311055867435, −1.59052966698578454412565905841, 0.867270595094514557535516408804, 3.40721570318424876485693636567, 4.48016911064177278761383194133, 5.31773479230242684593125158768, 5.91823677984193557161147877963, 7.35728982935419878668986486692, 8.389028515568925348863213922149, 8.807560769498729393505357952329, 9.550384919587653773488963561246, 10.77924728486757372646088730278

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