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Properties

Label 2-552-8.5-c1-0-42
Degree $2$
Conductor $552$
Sign $-0.873 - 0.487i$
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  + (0.238 − 1.39i)2-s + i·3-s + (−1.88 − 0.665i)4-s − 4.08i·5-s + (1.39 + 0.238i)6-s − 1.92·7-s + (−1.37 + 2.46i)8-s − 9-s + (−5.68 − 0.974i)10-s − 3.31i·11-s + (0.665 − 1.88i)12-s + 3.23i·13-s + (−0.459 + 2.68i)14-s + 4.08·15-s + (3.11 + 2.51i)16-s − 5.13·17-s + ⋯
L(s)  = 1  + (0.168 − 0.985i)2-s + 0.577i·3-s + (−0.942 − 0.332i)4-s − 1.82i·5-s + (0.569 + 0.0975i)6-s − 0.727·7-s + (−0.487 + 0.873i)8-s − 0.333·9-s + (−1.79 − 0.308i)10-s − 1.00i·11-s + (0.192 − 0.544i)12-s + 0.897i·13-s + (−0.122 + 0.717i)14-s + 1.05·15-s + (0.778 + 0.627i)16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.873 - 0.487i$
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: \((2,\ 552,\ (\ :1/2),\ -0.873 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164534 + 0.632363i\)
\(L(\frac12)\) \(\approx\) \(0.164534 + 0.632363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.238 + 1.39i)T \)
3 \( 1 - iT \)
23 \( 1 - T \)
good5 \( 1 + 4.08iT - 5T^{2} \)
7 \( 1 + 1.92T + 7T^{2} \)
11 \( 1 + 3.31iT - 11T^{2} \)
13 \( 1 - 3.23iT - 13T^{2} \)
17 \( 1 + 5.13T + 17T^{2} \)
19 \( 1 - 5.11iT - 19T^{2} \)
29 \( 1 + 7.41iT - 29T^{2} \)
31 \( 1 + 1.83T + 31T^{2} \)
37 \( 1 + 7.80iT - 37T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + 2.04iT - 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 3.69iT - 53T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 + 3.89iT - 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 - 1.12T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 3.52T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + 5.70T + 89T^{2} \)
97 \( 1 - 0.894T + 97T^{2} \)
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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

−10.21512002978586616458874751354, −9.296855946292963912897156406123, −8.931595927504299742810720774008, −8.146177553823686704403589274660, −6.16343444984359407550223350748, −5.32060898977202171840081581409, −4.32221120385564655805048042435, −3.67101770089546097613819392679, −1.96589531563786065501509003921, −0.34341619904447280788475888307, 2.60306886520733550113560155301, 3.49397402311085507614880379731, 4.98240341852703938899659068229, 6.25423691569315101370489275172, 6.96884373912237314241904583089, 7.14731709932489901996885629113, 8.391702765579931327770311998112, 9.502465898237768876557876292015, 10.34396706686054554588693916290, 11.19860892785867241125863164324

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