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Properties

Label 2-552-184.91-c1-0-43
Degree $2$
Conductor $552$
Sign $0.766 + 0.642i$
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  + (1.34 − 0.422i)2-s + 3-s + (1.64 − 1.14i)4-s − 1.97·5-s + (1.34 − 0.422i)6-s + 2.69·7-s + (1.73 − 2.23i)8-s + 9-s + (−2.66 + 0.834i)10-s − 0.719i·11-s + (1.64 − 1.14i)12-s − 0.585i·13-s + (3.63 − 1.13i)14-s − 1.97·15-s + (1.39 − 3.74i)16-s + 5.15i·17-s + ⋯
L(s)  = 1  + (0.954 − 0.299i)2-s + 0.577·3-s + (0.821 − 0.570i)4-s − 0.882·5-s + (0.550 − 0.172i)6-s + 1.01·7-s + (0.612 − 0.790i)8-s + 0.333·9-s + (−0.842 + 0.263i)10-s − 0.217i·11-s + (0.474 − 0.329i)12-s − 0.162i·13-s + (0.971 − 0.304i)14-s − 0.509·15-s + (0.348 − 0.937i)16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79066 - 1.01460i\)
\(L(\frac12)\) \(\approx\) \(2.79066 - 1.01460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.422i)T \)
3 \( 1 - T \)
23 \( 1 + (0.180 - 4.79i)T \)
good5 \( 1 + 1.97T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 + 0.719iT - 11T^{2} \)
13 \( 1 + 0.585iT - 13T^{2} \)
17 \( 1 - 5.15iT - 17T^{2} \)
19 \( 1 + 6.29iT - 19T^{2} \)
29 \( 1 - 1.33iT - 29T^{2} \)
31 \( 1 + 0.359iT - 31T^{2} \)
37 \( 1 - 3.15T + 37T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 + 0.124iT - 43T^{2} \)
47 \( 1 - 7.12iT - 47T^{2} \)
53 \( 1 - 2.15T + 53T^{2} \)
59 \( 1 + 7.69T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 - 8.35iT - 71T^{2} \)
73 \( 1 + 0.472T + 73T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 - 9.57iT - 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 - 7.32iT - 97T^{2} \)
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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

−11.12583605252016098753954236020, −10.02261032942734960263439506483, −8.783211453674921623865088795294, −7.87801276032988914852450472185, −7.18935867346856248846174156722, −5.90761307610108744084966520083, −4.75230631675683678247914291028, −4.00202019574133831089545754450, −2.94817426964597972385727330013, −1.55003231807090067679984658360, 1.94984305073569077266412434125, 3.28175282749425885413208225290, 4.29140513093107588062058871035, 4.99507991913166720784847077189, 6.31400412013110048890552631354, 7.52765372213540761805625110311, 7.86439551468581515278940149931, 8.789353161035148036533601100527, 10.17004018971933077013809744724, 11.18584532390881963251905430747

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