Properties

Label 2-552-184.101-c1-0-29
Degree $2$
Conductor $552$
Sign $0.999 - 0.0385i$
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  + (−0.674 + 1.24i)2-s + (0.755 − 0.654i)3-s + (−1.08 − 1.67i)4-s + (3.34 − 0.480i)5-s + (0.303 + 1.38i)6-s + (−0.0697 + 0.152i)7-s + (2.81 − 0.221i)8-s + (0.142 − 0.989i)9-s + (−1.65 + 4.47i)10-s + (0.816 − 2.77i)11-s + (−1.92 − 0.554i)12-s + (0.257 − 0.117i)13-s + (−0.142 − 0.189i)14-s + (2.21 − 2.55i)15-s + (−1.62 + 3.65i)16-s + (−0.333 − 0.214i)17-s + ⋯
L(s)  = 1  + (−0.477 + 0.878i)2-s + (0.436 − 0.378i)3-s + (−0.544 − 0.838i)4-s + (1.49 − 0.214i)5-s + (0.124 + 0.563i)6-s + (−0.0263 + 0.0577i)7-s + (0.996 − 0.0783i)8-s + (0.0474 − 0.329i)9-s + (−0.524 + 1.41i)10-s + (0.246 − 0.837i)11-s + (−0.554 − 0.160i)12-s + (0.0715 − 0.0326i)13-s + (−0.0381 − 0.0507i)14-s + (0.570 − 0.658i)15-s + (−0.406 + 0.913i)16-s + (−0.0809 − 0.0519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.999 - 0.0385i$
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: \((2,\ 552,\ (\ :1/2),\ 0.999 - 0.0385i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60788 + 0.0309795i\)
\(L(\frac12)\) \(\approx\) \(1.60788 + 0.0309795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.674 - 1.24i)T \)
3 \( 1 + (-0.755 + 0.654i)T \)
23 \( 1 + (3.99 + 2.66i)T \)
good5 \( 1 + (-3.34 + 0.480i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.0697 - 0.152i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.816 + 2.77i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.257 + 0.117i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.333 + 0.214i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (0.495 + 0.770i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (4.21 - 6.56i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-7.04 + 8.13i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-8.61 - 1.23i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.340 + 2.36i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.64 - 6.62i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 5.97T + 47T^{2} \)
53 \( 1 + (-4.46 - 2.04i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (5.23 - 2.39i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-4.46 - 3.87i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-3.11 - 10.6i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (4.45 - 1.30i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (11.0 - 7.09i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (1.02 + 2.25i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (9.74 + 1.40i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (1.33 + 1.54i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.942 + 6.55i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.37357464798116581965677997088, −9.685009102118630080291735416005, −8.929575421724672997107520146597, −8.273919046435631737948176941221, −7.15587406600929869349296195352, −6.09201965995356941799748676351, −5.77107273713836144377330577892, −4.36464880266533623302161423558, −2.52262898705358580488490763482, −1.19735487469277655631381549298, 1.70936647356890646615433876495, 2.52891614234975056051332104560, 3.84308738273668781349386719196, 4.95721316243466801534942320598, 6.19429372465892355542580017198, 7.38280931403545054387144857243, 8.465470218090410929901324028396, 9.357105715202612529594574216524, 9.974170231368009297657849344360, 10.35274070628940406984598074990

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