↑

Properties

Label 2-552-23.9-c1-0-7
Degree $2$
Conductor $552$
Sign $0.961 + 0.276i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.654 − 0.755i)3-s + (0.309 + 2.14i)5-s + (1.22 − 2.68i)7-s + (−0.142 + 0.989i)9-s + (0.937 + 0.275i)11-s + (−0.279 − 0.611i)13-s + (1.42 − 1.64i)15-s + (2.67 + 1.72i)17-s + (1.94 − 1.25i)19-s + (−2.83 + 0.831i)21-s + (4.76 + 0.546i)23-s + (0.270 − 0.0795i)25-s + (0.841 − 0.540i)27-s + (0.860 + 0.552i)29-s + (2.27 − 2.62i)31-s + ⋯
L(s)  = 1  + (−0.378 − 0.436i)3-s + (0.138 + 0.961i)5-s + (0.463 − 1.01i)7-s + (−0.0474 + 0.329i)9-s + (0.282 + 0.0830i)11-s + (−0.0774 − 0.169i)13-s + (0.367 − 0.423i)15-s + (0.649 + 0.417i)17-s + (0.446 − 0.287i)19-s + (−0.617 + 0.181i)21-s + (0.993 + 0.114i)23-s + (0.0541 − 0.0159i)25-s + (0.161 − 0.104i)27-s + (0.159 + 0.102i)29-s + (0.408 − 0.471i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42018 - 0.200258i\)
\(L(\frac12)\) \(\approx\) \(1.42018 - 0.200258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-4.76 - 0.546i)T \)
good5 \( 1 + (-0.309 - 2.14i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-1.22 + 2.68i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.937 - 0.275i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.279 + 0.611i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-2.67 - 1.72i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-1.94 + 1.25i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-0.860 - 0.552i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-2.27 + 2.62i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.116 - 0.813i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.840 + 5.84i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (0.0622 + 0.0718i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + (0.944 - 2.06i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (2.17 + 4.76i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-1.64 + 1.89i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (11.2 - 3.30i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (0.504 - 0.148i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (7.15 - 4.59i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (5.51 + 12.0i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.46 - 10.1i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (0.750 + 0.866i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.23 - 8.60i)T + (-93.0 + 27.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68407469618146480820765904612, −10.24017877747070024850884666139, −8.991057003874320366553466881041, −7.71035125109279700824899089227, −7.19562993202427538352291402931, −6.34825546809860855036855494445, −5.23477840630220041709780025563, −4.00855745579209457978874101382, −2.76963036279105354709994429865, −1.15476916543169958494744063689, 1.26059292670423227345218248621, 2.93369343588132125503586288770, 4.44932908477892874849835586448, 5.22813088318247911902816952213, 5.91621762017920779578470045768, 7.24430049554683764211001783306, 8.478136566638809411981946964104, 9.023734671420894706217697549782, 9.811549327447127879442224648566, 10.88542074216828090711802922057

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