↑

Properties

Label 2-1550-31.25-c1-0-25
Degree $2$
Conductor $1550$
Sign $0.970 - 0.242i$
Analytic cond. $12.3768$
Root an. cond. $3.51806$
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  − 2-s + (1.5 + 2.59i)3-s + 4-s + (−1.5 − 2.59i)6-s + (−1.5 − 2.59i)7-s − 8-s + (−3 + 5.19i)9-s + (1.5 − 2.59i)11-s + (1.5 + 2.59i)12-s + (2.5 − 4.33i)13-s + (1.5 + 2.59i)14-s + 16-s + (1.5 + 2.59i)17-s + (3 − 5.19i)18-s + (−3.5 − 6.06i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.866 + 1.49i)3-s + 0.5·4-s + (−0.612 − 1.06i)6-s + (−0.566 − 0.981i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.452 − 0.783i)11-s + (0.433 + 0.749i)12-s + (0.693 − 1.20i)13-s + (0.400 + 0.694i)14-s + 0.250·16-s + (0.363 + 0.630i)17-s + (0.707 − 1.22i)18-s + (−0.802 − 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1550\)    =    \(2 \cdot 5^{2} \cdot 31\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(12.3768\)
Root analytic conductor: \(3.51806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1550} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1550,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.557388645\)
\(L(\frac12)\) \(\approx\) \(1.557388645\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
31 \( 1 + (-2 - 5.19i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.5 + 4.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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

−9.400532687095453127680890499248, −8.748572182046043934737968065037, −8.316115792887110502293531384250, −7.27467291218636123288388393236, −6.30824855979043637895948867889, −5.24255841094099411474680145174, −4.11909092469512741639154040748, −3.43474117563923714093981951373, −2.73475127062637814312660478101, −0.78741968891294778729276024837, 1.22720013501620624514225641620, 2.11074717237648254619311801577, 2.85927476941320107729774456271, 4.08146242983836378487279421890, 5.82305616963007530066873494108, 6.50166608988406819120841605408, 7.03396242678118190309843241181, 7.953143473278664439508058534772, 8.540756854176018731855620390925, 9.304035029967151177625657420809

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