↑

Properties

Label 2-552-23.2-c1-0-8
Degree $2$
Conductor $552$
Sign $0.219 + 0.975i$
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  + (−0.142 − 0.989i)3-s + (−0.451 − 0.132i)5-s + (−0.325 + 0.375i)7-s + (−0.959 + 0.281i)9-s + (3.66 − 2.35i)11-s + (1.44 + 1.66i)13-s + (−0.0669 + 0.465i)15-s + (2.21 − 4.85i)17-s + (−2.87 − 6.30i)19-s + (0.417 + 0.268i)21-s + (4.54 − 1.52i)23-s + (−4.02 − 2.58i)25-s + (0.415 + 0.909i)27-s + (−2.79 + 6.12i)29-s + (1.52 − 10.5i)31-s + ⋯
L(s)  = 1  + (−0.0821 − 0.571i)3-s + (−0.201 − 0.0592i)5-s + (−0.122 + 0.141i)7-s + (−0.319 + 0.0939i)9-s + (1.10 − 0.709i)11-s + (0.400 + 0.462i)13-s + (−0.0172 + 0.120i)15-s + (0.537 − 1.17i)17-s + (−0.660 − 1.44i)19-s + (0.0911 + 0.0585i)21-s + (0.948 − 0.317i)23-s + (−0.804 − 0.516i)25-s + (0.0799 + 0.175i)27-s + (−0.519 + 1.13i)29-s + (0.273 − 1.89i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.219 + 0.975i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.219 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03617 - 0.829381i\)
\(L(\frac12)\) \(\approx\) \(1.03617 - 0.829381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-4.54 + 1.52i)T \)
good5 \( 1 + (0.451 + 0.132i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (0.325 - 0.375i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-3.66 + 2.35i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-1.44 - 1.66i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-2.21 + 4.85i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (2.87 + 6.30i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (2.79 - 6.12i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.52 + 10.5i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (4.47 - 1.31i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.87 - 0.549i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.727 - 5.06i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 4.15T + 47T^{2} \)
53 \( 1 + (-4.82 + 5.56i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (3.87 + 4.47i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-0.120 + 0.836i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (5.09 + 3.27i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-11.4 - 7.33i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.276 + 0.604i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-3.92 - 4.52i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-13.5 + 3.98i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-2.08 - 14.5i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-9.30 - 2.73i)T + (81.6 + 52.4i)T^{2} \)
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

−10.92846444579997063686371346668, −9.393053034247293901455010853191, −8.971018365277117818078029172206, −7.87558258846773454044257588651, −6.86827978277850263847915860461, −6.22546358700139381760559386038, −5.00914434488188729479363799963, −3.78934252464640893010732844186, −2.51580966058322416087458919441, −0.847518226565792258517189591252, 1.62863188776722813488739562395, 3.53618257571741344508704850681, 4.07602384631366397533948062229, 5.45496298809153008938999884810, 6.32000440085759436211624142428, 7.41794936775685177574523189662, 8.423948533970374914286910918336, 9.258721816644265107049447897210, 10.25049212762877307466762180301, 10.71579712856697127536807539554

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