↑

Properties

Label 2-550-11.3-c1-0-3
Degree $2$
Conductor $550$
Sign $-0.561 - 0.827i$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
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  + (−0.809 + 0.587i)2-s + (0.650 + 2.00i)3-s + (0.309 − 0.951i)4-s + (−1.70 − 1.23i)6-s + (0.675 − 2.07i)7-s + (0.309 + 0.951i)8-s + (−1.15 + 0.841i)9-s + (1.92 + 2.69i)11-s + 2.10·12-s + (−5.11 + 3.71i)13-s + (0.675 + 2.07i)14-s + (−0.809 − 0.587i)16-s + (1.34 + 0.980i)17-s + (0.442 − 1.36i)18-s + (1.82 + 5.60i)19-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.375 + 1.15i)3-s + (0.154 − 0.475i)4-s + (−0.695 − 0.505i)6-s + (0.255 − 0.785i)7-s + (0.109 + 0.336i)8-s + (−0.386 + 0.280i)9-s + (0.581 + 0.813i)11-s + 0.607·12-s + (−1.41 + 1.03i)13-s + (0.180 + 0.555i)14-s + (−0.202 − 0.146i)16-s + (0.327 + 0.237i)17-s + (0.104 − 0.320i)18-s + (0.417 + 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ -0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557805 + 1.05263i\)
\(L(\frac12)\) \(\approx\) \(0.557805 + 1.05263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-1.92 - 2.69i)T \)
good3 \( 1 + (-0.650 - 2.00i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.675 + 2.07i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.11 - 3.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.34 - 0.980i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.82 - 5.60i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.26T + 23T^{2} \)
29 \( 1 + (1.95 - 6.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.74 - 2.72i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.849 + 2.61i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.89 + 8.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.29T + 43T^{2} \)
47 \( 1 + (0.582 + 1.79i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.36 + 4.62i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.47 + 7.63i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.51 - 2.55i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + (11.1 + 8.10i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.32 - 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.40 - 1.02i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.25 - 5.27i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-9.72 + 7.06i)T + (29.9 - 92.2i)T^{2} \)
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

−10.62074116614615831428645722937, −10.06449587464972092913424780105, −9.413460237694246293982684324282, −8.710084571123369228014191852062, −7.35371345332676824352463980707, −6.99606331656168775224528874901, −5.35278491314745978375729539211, −4.47291935167079687935222358407, −3.59316705458461500046740865596, −1.75255474248032739051245266165, 0.831261255819731931866888740491, 2.31857504574534269189189689471, 3.04748404304134674578333506743, 4.89868559367707503641232195070, 6.08044256273192285728790867419, 7.24814430861945074701661428347, 7.77296055558466941921142656990, 8.709810524315394074632931114422, 9.424587577136005897628664293945, 10.46273312124504642728511386559

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