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Properties

Label 2-552-184.91-c1-0-5
Degree $2$
Conductor $552$
Sign $-0.918 - 0.395i$
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  + (−0.695 + 1.23i)2-s + 3-s + (−1.03 − 1.71i)4-s − 1.12·5-s + (−0.695 + 1.23i)6-s − 2.04·7-s + (2.82 − 0.0816i)8-s + 9-s + (0.783 − 1.38i)10-s + 3.16i·11-s + (−1.03 − 1.71i)12-s + 0.675i·13-s + (1.41 − 2.51i)14-s − 1.12·15-s + (−1.86 + 3.53i)16-s + 7.33i·17-s + ⋯
L(s)  = 1  + (−0.491 + 0.870i)2-s + 0.577·3-s + (−0.516 − 0.856i)4-s − 0.503·5-s + (−0.283 + 0.502i)6-s − 0.771·7-s + (0.999 − 0.0288i)8-s + 0.333·9-s + (0.247 − 0.438i)10-s + 0.955i·11-s + (−0.298 − 0.494i)12-s + 0.187i·13-s + (0.379 − 0.671i)14-s − 0.290·15-s + (−0.466 + 0.884i)16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.918 - 0.395i$
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: \((2,\ 552,\ (\ :1/2),\ -0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146914 + 0.713313i\)
\(L(\frac12)\) \(\approx\) \(0.146914 + 0.713313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.695 - 1.23i)T \)
3 \( 1 - T \)
23 \( 1 + (4.34 + 2.02i)T \)
good5 \( 1 + 1.12T + 5T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 - 3.16iT - 11T^{2} \)
13 \( 1 - 0.675iT - 13T^{2} \)
17 \( 1 - 7.33iT - 17T^{2} \)
19 \( 1 + 0.636iT - 19T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 - 8.08iT - 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 0.447iT - 43T^{2} \)
47 \( 1 - 4.96iT - 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 - 0.726T + 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 + 0.0830iT - 71T^{2} \)
73 \( 1 - 1.54T + 73T^{2} \)
79 \( 1 + 6.84T + 79T^{2} \)
83 \( 1 + 8.66iT - 83T^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 - 0.322iT - 97T^{2} \)
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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

−10.71510640395122841762502947739, −10.12258630764628081458532932568, −9.232151596545091414700458302918, −8.452072071689837370596596434779, −7.64241070349709077027031213523, −6.80996345592329579360575538166, −5.95254377246030922970920852467, −4.55711239197697948055309882432, −3.64688094931056239352863076378, −1.82323530677620756856901561743, 0.45452074874871685913004798838, 2.41302538772647602354666420351, 3.37415994465050104562207355654, 4.18908151596009826459172445437, 5.71950151026862374106301111842, 7.20143945802757669113774061646, 7.88771247600314172136882968135, 8.813020825342302652083878218813, 9.579600448482839050944245753992, 10.21850369764612063066351595588

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