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Properties

Label 2-552-184.91-c1-0-21
Degree $2$
Conductor $552$
Sign $0.335 - 0.942i$
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  + (1.07 + 0.914i)2-s − 3-s + (0.327 + 1.97i)4-s + 0.969·5-s + (−1.07 − 0.914i)6-s + 4.55·7-s + (−1.45 + 2.42i)8-s + 9-s + (1.04 + 0.886i)10-s − 0.915i·11-s + (−0.327 − 1.97i)12-s − 4.49i·13-s + (4.91 + 4.16i)14-s − 0.969·15-s + (−3.78 + 1.29i)16-s + 5.37i·17-s + ⋯
L(s)  = 1  + (0.762 + 0.646i)2-s − 0.577·3-s + (0.163 + 0.986i)4-s + 0.433·5-s + (−0.440 − 0.373i)6-s + 1.72·7-s + (−0.513 + 0.858i)8-s + 0.333·9-s + (0.330 + 0.280i)10-s − 0.276i·11-s + (−0.0944 − 0.569i)12-s − 1.24i·13-s + (1.31 + 1.11i)14-s − 0.250·15-s + (−0.946 + 0.322i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.335 - 0.942i$
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: \((2,\ 552,\ (\ :1/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84693 + 1.30276i\)
\(L(\frac12)\) \(\approx\) \(1.84693 + 1.30276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 - 0.914i)T \)
3 \( 1 + T \)
23 \( 1 + (-4.70 + 0.937i)T \)
good5 \( 1 - 0.969T + 5T^{2} \)
7 \( 1 - 4.55T + 7T^{2} \)
11 \( 1 + 0.915iT - 11T^{2} \)
13 \( 1 + 4.49iT - 13T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + 0.688iT - 19T^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 - 5.64iT - 31T^{2} \)
37 \( 1 - 3.25T + 37T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 + 9.54iT - 43T^{2} \)
47 \( 1 - 0.156iT - 47T^{2} \)
53 \( 1 + 2.42T + 53T^{2} \)
59 \( 1 - 3.81T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + 8.18T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + 17.8iT - 97T^{2} \)
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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

−10.92809094810186884522842280869, −10.53547048367413266425574484966, −8.810663289348982984808924988964, −8.140940561939992932241243060205, −7.30087537185412046835634352236, −6.14869564735571717805892644887, −5.31335747580195636984929428277, −4.76731934653766506900521258760, −3.40460697625821036603479704700, −1.72955678998440822940271841342, 1.37505016837529767434357951767, 2.38552531313115233810826924573, 4.23915666422915899069295500479, 4.83478513815118037414467562971, 5.68623952869349984454677995232, 6.76946098303243757508927932851, 7.78994144540811362411616354012, 9.206855697525950469288377928029, 9.885647882672828567058238034041, 11.00473209452695582573827670000

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