Properties

Label 2-552-184.13-c1-0-6
Degree $2$
Conductor $552$
Sign $-0.600 - 0.799i$
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  + (−0.464 + 1.33i)2-s + (−0.540 + 0.841i)3-s + (−1.56 − 1.24i)4-s + (0.798 + 0.364i)5-s + (−0.872 − 1.11i)6-s + (2.23 + 0.657i)7-s + (2.38 − 1.51i)8-s + (−0.415 − 0.909i)9-s + (−0.858 + 0.897i)10-s + (−0.502 − 0.435i)11-s + (1.89 − 0.648i)12-s + (0.537 + 1.82i)13-s + (−1.91 + 2.68i)14-s + (−0.738 + 0.474i)15-s + (0.921 + 3.89i)16-s + (0.515 + 3.58i)17-s + ⋯
L(s)  = 1  + (−0.328 + 0.944i)2-s + (−0.312 + 0.485i)3-s + (−0.784 − 0.620i)4-s + (0.357 + 0.163i)5-s + (−0.356 − 0.454i)6-s + (0.846 + 0.248i)7-s + (0.843 − 0.537i)8-s + (−0.138 − 0.303i)9-s + (−0.271 + 0.283i)10-s + (−0.151 − 0.131i)11-s + (0.546 − 0.187i)12-s + (0.148 + 0.507i)13-s + (−0.512 + 0.717i)14-s + (−0.190 + 0.122i)15-s + (0.230 + 0.973i)16-s + (0.125 + 0.870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515663 + 1.03291i\)
\(L(\frac12)\) \(\approx\) \(0.515663 + 1.03291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.464 - 1.33i)T \)
3 \( 1 + (0.540 - 0.841i)T \)
23 \( 1 + (-2.85 - 3.85i)T \)
good5 \( 1 + (-0.798 - 0.364i)T + (3.27 + 3.77i)T^{2} \)
7 \( 1 + (-2.23 - 0.657i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.502 + 0.435i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.537 - 1.82i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.515 - 3.58i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-7.17 - 1.03i)T + (18.2 + 5.35i)T^{2} \)
29 \( 1 + (7.91 - 1.13i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-0.262 + 0.168i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-5.02 + 2.29i)T + (24.2 - 27.9i)T^{2} \)
41 \( 1 + (4.68 - 10.2i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (4.26 - 6.63i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + 1.07T + 47T^{2} \)
53 \( 1 + (-1.94 + 6.62i)T + (-44.5 - 28.6i)T^{2} \)
59 \( 1 + (-1.87 - 6.38i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (3.59 + 5.59i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-11.3 + 9.87i)T + (9.53 - 66.3i)T^{2} \)
71 \( 1 + (-0.962 - 1.11i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.90 - 13.2i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-7.03 + 2.06i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-2.45 + 1.12i)T + (54.3 - 62.7i)T^{2} \)
89 \( 1 + (-6.06 - 3.90i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (2.94 - 6.43i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.05986754576494085499552954584, −9.872057536031725870134784144964, −9.452361963717743978221792962973, −8.312486766898108076025530483598, −7.62793235255399924031053262935, −6.45143708825701624499449778575, −5.57586839554267352828198331642, −4.89330126871645278296678651889, −3.64278675288513500556559417703, −1.54183095473503405347657473409, 0.880626669844135234137564240820, 2.10963517954203843504394074285, 3.42405204521231561653408510467, 4.89078500812483399932427454756, 5.50665395188123836354467363444, 7.23997667574482402617244773156, 7.77993797544189000364803114273, 8.877188435786127419520297121217, 9.688025598886332834011771692965, 10.61295301795618167050241683691

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