↑

Properties

Label 2-552-23.3-c1-0-6
Degree $2$
Conductor $552$
Sign $0.999 - 0.0167i$
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  + (0.415 − 0.909i)3-s + (−1.96 + 2.26i)5-s + (3.59 − 2.30i)7-s + (−0.654 − 0.755i)9-s + (−0.492 + 3.42i)11-s + (2.61 + 1.67i)13-s + (1.24 + 2.73i)15-s + (−0.416 − 0.122i)17-s + (5.69 − 1.67i)19-s + (−0.608 − 4.22i)21-s + (1.75 − 4.46i)23-s + (−0.571 − 3.97i)25-s + (−0.959 + 0.281i)27-s + (5.56 + 1.63i)29-s + (2.51 + 5.51i)31-s + ⋯
L(s)  = 1  + (0.239 − 0.525i)3-s + (−0.879 + 1.01i)5-s + (1.35 − 0.873i)7-s + (−0.218 − 0.251i)9-s + (−0.148 + 1.03i)11-s + (0.724 + 0.465i)13-s + (0.322 + 0.705i)15-s + (−0.101 − 0.0296i)17-s + (1.30 − 0.383i)19-s + (−0.132 − 0.922i)21-s + (0.366 − 0.930i)23-s + (−0.114 − 0.794i)25-s + (−0.184 + 0.0542i)27-s + (1.03 + 0.303i)29-s + (0.452 + 0.990i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.999 - 0.0167i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.999 - 0.0167i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60630 + 0.0134786i\)
\(L(\frac12)\) \(\approx\) \(1.60630 + 0.0134786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-1.75 + 4.46i)T \)
good5 \( 1 + (1.96 - 2.26i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-3.59 + 2.30i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.492 - 3.42i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-2.61 - 1.67i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.416 + 0.122i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-5.69 + 1.67i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-5.56 - 1.63i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-2.51 - 5.51i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (2.84 + 3.28i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-4.31 + 4.97i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.59 - 7.88i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 2.34T + 47T^{2} \)
53 \( 1 + (-2.29 + 1.47i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-3.95 - 2.54i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (4.40 + 9.65i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-1.72 - 12.0i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (0.995 + 6.92i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (8.86 - 2.60i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (1.94 + 1.24i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (11.7 + 13.5i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (3.49 - 7.65i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (11.6 - 13.4i)T + (-13.8 - 96.0i)T^{2} \)
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

−10.93267219588175382614169949651, −10.16076886319755791517969775014, −8.768626240623499670776867847347, −7.897893550369171819174419852522, −7.22151290000559571299127438816, −6.70486082143554020388034272597, −4.97151248400772136251551645738, −4.07888078334625410537530180308, −2.85951250936831159951251974450, −1.35172213971060998544876128057, 1.19730930461009165571551381964, 3.03604481055339890899247020973, 4.18983394847995982716126417887, 5.16012199476758243725907806601, 5.77257223435216696576120766965, 7.65231720092819256878078625027, 8.404658878948519114332820306121, 8.627265984986968747657176066114, 9.787439256570340330273501346224, 11.05638236905339989660720744597

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