Properties

Label 2-552-8.5-c1-0-35
Degree $2$
Conductor $552$
Sign $0.871 + 0.491i$
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  + (1.08 + 0.905i)2-s i·3-s + (0.360 + 1.96i)4-s − 3.71i·5-s + (0.905 − 1.08i)6-s + 1.13·7-s + (−1.38 + 2.46i)8-s − 9-s + (3.36 − 4.03i)10-s + 1.53i·11-s + (1.96 − 0.360i)12-s − 4.95i·13-s + (1.23 + 1.02i)14-s − 3.71·15-s + (−3.73 + 1.41i)16-s + 7.22·17-s + ⋯
L(s)  = 1  + (0.768 + 0.640i)2-s − 0.577i·3-s + (0.180 + 0.983i)4-s − 1.66i·5-s + (0.369 − 0.443i)6-s + 0.429·7-s + (−0.491 + 0.871i)8-s − 0.333·9-s + (1.06 − 1.27i)10-s + 0.463i·11-s + (0.567 − 0.104i)12-s − 1.37i·13-s + (0.330 + 0.275i)14-s − 0.959·15-s + (−0.934 + 0.354i)16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.871 + 0.491i$
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: \((2,\ 552,\ (\ :1/2),\ 0.871 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15982 - 0.566768i\)
\(L(\frac12)\) \(\approx\) \(2.15982 - 0.566768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.905i)T \)
3 \( 1 + iT \)
23 \( 1 - T \)
good5 \( 1 + 3.71iT - 5T^{2} \)
7 \( 1 - 1.13T + 7T^{2} \)
11 \( 1 - 1.53iT - 11T^{2} \)
13 \( 1 + 4.95iT - 13T^{2} \)
17 \( 1 - 7.22T + 17T^{2} \)
19 \( 1 + 4.69iT - 19T^{2} \)
29 \( 1 - 2.85iT - 29T^{2} \)
31 \( 1 - 0.770T + 31T^{2} \)
37 \( 1 + 6.52iT - 37T^{2} \)
41 \( 1 + 3.64T + 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 - 6.27T + 47T^{2} \)
53 \( 1 - 14.2iT - 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 + 2.32iT - 61T^{2} \)
67 \( 1 - 6.38iT - 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 1.06T + 73T^{2} \)
79 \( 1 - 1.16T + 79T^{2} \)
83 \( 1 - 8.10iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 6.44T + 97T^{2} \)
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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

−10.98490292734818422309241443700, −9.560072755234582987200185548955, −8.585303235626451744548007909413, −7.929350620825303774028711638669, −7.26279824384983807824938620215, −5.74958158608666920016950027419, −5.27829219309983695356839404044, −4.36752887514451720161033798336, −2.91277408257629121907742397390, −1.13751427043507403846185834568, 1.94881213559745427135664906848, 3.25833368803192943742350096793, 3.82496140639418690290753679989, 5.18021287721475594052421692487, 6.15340530099657662579953245451, 6.95587727527232243833720704362, 8.173097069786320686200734499055, 9.665533551258577244243935201904, 10.13399308981957956124536722364, 10.95645621568797695468907897012

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