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Properties

Label 2-552-184.101-c1-0-32
Degree $2$
Conductor $552$
Sign $-0.801 + 0.597i$
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  + (−0.351 − 1.36i)2-s + (0.755 − 0.654i)3-s + (−1.75 + 0.962i)4-s + (−0.195 + 0.0280i)5-s + (−1.16 − 0.805i)6-s + (−0.184 + 0.404i)7-s + (1.93 + 2.06i)8-s + (0.142 − 0.989i)9-s + (0.107 + 0.257i)10-s + (1.13 − 3.85i)11-s + (−0.694 + 1.87i)12-s + (3.47 − 1.58i)13-s + (0.618 + 0.110i)14-s + (−0.129 + 0.149i)15-s + (2.14 − 3.37i)16-s + (−5.85 − 3.76i)17-s + ⋯
L(s)  = 1  + (−0.248 − 0.968i)2-s + (0.436 − 0.378i)3-s + (−0.876 + 0.481i)4-s + (−0.0873 + 0.0125i)5-s + (−0.474 − 0.328i)6-s + (−0.0698 + 0.152i)7-s + (0.684 + 0.729i)8-s + (0.0474 − 0.329i)9-s + (0.0338 + 0.0815i)10-s + (0.341 − 1.16i)11-s + (−0.200 + 0.541i)12-s + (0.964 − 0.440i)13-s + (0.165 + 0.0296i)14-s + (−0.0333 + 0.0385i)15-s + (0.536 − 0.843i)16-s + (−1.42 − 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.801 + 0.597i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.801 + 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379520 - 1.14469i\)
\(L(\frac12)\) \(\approx\) \(0.379520 - 1.14469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.351 + 1.36i)T \)
3 \( 1 + (-0.755 + 0.654i)T \)
23 \( 1 + (-2.94 + 3.78i)T \)
good5 \( 1 + (0.195 - 0.0280i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.184 - 0.404i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-1.13 + 3.85i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-3.47 + 1.58i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (5.85 + 3.76i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (0.936 + 1.45i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-1.09 + 1.70i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (1.09 - 1.26i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-1.63 - 0.235i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.224 + 1.56i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (1.44 - 1.24i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 6.84T + 47T^{2} \)
53 \( 1 + (-3.92 - 1.79i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-10.5 + 4.81i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (6.00 + 5.20i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (2.42 + 8.25i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (4.95 - 1.45i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (10.6 - 6.86i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-6.34 - 13.8i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-12.6 - 1.82i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (5.55 + 6.41i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.0567 - 0.394i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.69896431549610972594245028344, −9.378393020192011117163851950896, −8.789575420067615154174904115821, −8.155008160566837934231963001628, −6.93559207332053426709768484985, −5.78229045488565695573310023126, −4.38673696563056419308946894351, −3.34201976352248477593001570755, −2.35997634189731757214408174246, −0.75680054177368653500232424177, 1.79941311995900346813697191861, 3.88772323993651290152693213850, 4.42735986653172842812062767527, 5.76801016323992427755366208356, 6.72876927232750319443022437358, 7.52437232868184383154814862844, 8.606758954834913301992584269595, 9.104517857715382475789154704978, 10.06217501895378973676685475519, 10.80599578935130787619714926032

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