Properties

Label 2-552-184.101-c1-0-39
Degree $2$
Conductor $552$
Sign $-0.244 + 0.969i$
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  + (1.19 − 0.757i)2-s + (−0.755 + 0.654i)3-s + (0.853 − 1.80i)4-s + (−2.01 + 0.290i)5-s + (−0.406 + 1.35i)6-s + (0.214 − 0.470i)7-s + (−0.349 − 2.80i)8-s + (0.142 − 0.989i)9-s + (−2.19 + 1.87i)10-s + (0.754 − 2.56i)11-s + (0.539 + 1.92i)12-s + (5.11 − 2.33i)13-s + (−0.0994 − 0.723i)14-s + (1.33 − 1.54i)15-s + (−2.54 − 3.08i)16-s + (−3.23 − 2.07i)17-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−0.436 + 0.378i)3-s + (0.426 − 0.904i)4-s + (−0.902 + 0.129i)5-s + (−0.166 + 0.552i)6-s + (0.0811 − 0.177i)7-s + (−0.123 − 0.992i)8-s + (0.0474 − 0.329i)9-s + (−0.693 + 0.593i)10-s + (0.227 − 0.774i)11-s + (0.155 + 0.555i)12-s + (1.41 − 0.647i)13-s + (−0.0265 − 0.193i)14-s + (0.344 − 0.398i)15-s + (−0.635 − 0.771i)16-s + (−0.784 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.244 + 0.969i$
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: \((2,\ 552,\ (\ :1/2),\ -0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999393 - 1.28307i\)
\(L(\frac12)\) \(\approx\) \(0.999393 - 1.28307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.757i)T \)
3 \( 1 + (0.755 - 0.654i)T \)
23 \( 1 + (2.90 + 3.81i)T \)
good5 \( 1 + (2.01 - 0.290i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.214 + 0.470i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.754 + 2.56i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-5.11 + 2.33i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (3.23 + 2.07i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (0.853 + 1.32i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-1.80 + 2.80i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.62 + 3.02i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-6.67 - 0.960i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (-0.979 - 6.81i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.51 - 6.50i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 7.60T + 47T^{2} \)
53 \( 1 + (-1.29 - 0.592i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (-0.925 + 0.422i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-4.46 - 3.87i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-0.456 - 1.55i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (14.2 - 4.17i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-10.3 + 6.62i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (3.77 + 8.27i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (5.36 + 0.770i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-8.92 - 10.3i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.0860 + 0.598i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.91114563584055396742339554942, −10.02548742384133772720205653216, −8.832446298048733888201629613238, −7.83327986956967015851063504713, −6.46876008810809761707172333839, −5.90880288238686708016801963680, −4.54463786122056574933305029196, −3.92336472003812302677864864053, −2.85031936015593467139377463617, −0.78138007725372602549412245749, 1.93663957147117728274204132181, 3.74035803055515161817166834858, 4.33443195866118155777520615738, 5.57393401506599525977973892981, 6.49053478912925792395246694796, 7.22636851643099738917275428619, 8.201319361314873228595145350280, 8.874843947110382615038972499060, 10.48959857961111671587091441514, 11.46843227320427311849640942487

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