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Properties

Label 2-552-23.4-c1-0-5
Degree $2$
Conductor $552$
Sign $0.247 + 0.968i$
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  + (−0.959 + 0.281i)3-s + (−2.59 − 1.66i)5-s + (0.565 + 3.93i)7-s + (0.841 − 0.540i)9-s + (0.839 − 1.83i)11-s + (0.139 − 0.966i)13-s + (2.95 + 0.867i)15-s + (−1.72 − 1.99i)17-s + (5.61 − 6.48i)19-s + (−1.65 − 3.61i)21-s + (1.39 − 4.58i)23-s + (1.86 + 4.07i)25-s + (−0.654 + 0.755i)27-s + (−3.80 − 4.39i)29-s + (−7.40 − 2.17i)31-s + ⋯
L(s)  = 1  + (−0.553 + 0.162i)3-s + (−1.15 − 0.744i)5-s + (0.213 + 1.48i)7-s + (0.280 − 0.180i)9-s + (0.253 − 0.554i)11-s + (0.0385 − 0.268i)13-s + (0.762 + 0.224i)15-s + (−0.419 − 0.483i)17-s + (1.28 − 1.48i)19-s + (−0.360 − 0.789i)21-s + (0.291 − 0.956i)23-s + (0.372 + 0.815i)25-s + (−0.126 + 0.145i)27-s + (−0.706 − 0.815i)29-s + (−1.32 − 0.390i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622040 - 0.483173i\)
\(L(\frac12)\) \(\approx\) \(0.622040 - 0.483173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-1.39 + 4.58i)T \)
good5 \( 1 + (2.59 + 1.66i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.565 - 3.93i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.839 + 1.83i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.139 + 0.966i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.72 + 1.99i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-5.61 + 6.48i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (3.80 + 4.39i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (7.40 + 2.17i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-2.96 + 1.90i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (1.82 + 1.17i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.70 + 0.499i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + (-0.691 - 4.81i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.34 + 9.38i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (4.20 + 1.23i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-3.34 - 7.33i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-4.28 - 9.37i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-10.4 + 12.1i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (1.14 - 7.99i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-1.37 + 0.883i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (6.72 - 1.97i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (7.33 + 4.71i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.09652933412136764774115959153, −9.378016362294621370327599003757, −8.956215837235541318676892501742, −8.033592353714490336855695875438, −7.03200428435742251032375738232, −5.71420463211958177947434072651, −5.07967417116505713221776690360, −4.03176877181803601015534196865, −2.63074697257691347209018777684, −0.53952426088016220632249278330, 1.39212612028186265570748490372, 3.58508005040890432775825760329, 4.04706266322236754189011282434, 5.38019092378783541899664744953, 6.74521380830583708449091388280, 7.42103250068170227285649316555, 7.80412950874602919706154389852, 9.366364644420874333868158629634, 10.43045276922763695182343086347, 10.94777912570840203015795201264

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