↑

Properties

Label 2-552-23.16-c1-0-8
Degree $2$
Conductor $552$
Sign $0.309 + 0.950i$
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  + (−0.841 + 0.540i)3-s + (−0.380 − 0.832i)5-s + (2.32 − 0.682i)7-s + (0.415 − 0.909i)9-s + (−2.20 − 2.54i)11-s + (−3.87 − 1.13i)13-s + (0.770 + 0.494i)15-s + (0.875 − 6.09i)17-s + (0.929 + 6.46i)19-s + (−1.58 + 1.83i)21-s + (4.54 − 1.53i)23-s + (2.72 − 3.14i)25-s + (0.142 + 0.989i)27-s + (0.505 − 3.51i)29-s + (−3.45 − 2.22i)31-s + ⋯
L(s)  = 1  + (−0.485 + 0.312i)3-s + (−0.170 − 0.372i)5-s + (0.878 − 0.257i)7-s + (0.138 − 0.303i)9-s + (−0.666 − 0.768i)11-s + (−1.07 − 0.315i)13-s + (0.198 + 0.127i)15-s + (0.212 − 1.47i)17-s + (0.213 + 1.48i)19-s + (−0.346 + 0.399i)21-s + (0.947 − 0.320i)23-s + (0.545 − 0.629i)25-s + (0.0273 + 0.190i)27-s + (0.0939 − 0.653i)29-s + (−0.620 − 0.398i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863459 - 0.627031i\)
\(L(\frac12)\) \(\approx\) \(0.863459 - 0.627031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-4.54 + 1.53i)T \)
good5 \( 1 + (0.380 + 0.832i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-2.32 + 0.682i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (2.20 + 2.54i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (3.87 + 1.13i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.875 + 6.09i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.929 - 6.46i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.505 + 3.51i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (3.45 + 2.22i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-3.49 + 7.66i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (4.08 + 8.94i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.12 + 3.29i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 3.53T + 47T^{2} \)
53 \( 1 + (9.76 - 2.86i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-8.04 - 2.36i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-7.43 - 4.77i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (2.85 - 3.29i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-2.05 + 2.36i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-2.20 - 15.3i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (11.4 + 3.36i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (4.10 - 8.99i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-2.70 + 1.73i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-3.96 - 8.68i)T + (-63.5 + 73.3i)T^{2} \)
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

−10.66161977873943192951807259461, −9.872343568298136596464329449899, −8.844317824906056727814889948382, −7.85580412797112380408494560446, −7.19067556562859051596540745028, −5.63778980169314810205271736779, −5.11547802461871151324430304354, −4.07703430601549344761884094220, −2.59927395200482392932140906773, −0.67530995777763123328279345537, 1.65701789401681714735851613590, 2.95019871821457810252130924982, 4.72218355568477400648209852212, 5.14810623152859350581687436335, 6.58526661890090360107821766467, 7.32279402628939327507301863433, 8.115642625255142533646165462420, 9.231525854901676169059378626418, 10.25439663938162327659526140649, 11.10044647493173429739437881965

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