↑

Properties

Label 2-55e2-275.213-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.193 + 0.981i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 1.39i)3-s + (−0.587 − 0.809i)4-s + (0.587 + 0.809i)5-s + 2.90i·9-s + (−0.309 + 1.95i)12-s + (0.309 − 1.95i)15-s + (−0.309 + 0.951i)16-s + (0.309 − 0.951i)20-s + (−0.142 − 0.896i)23-s + (−0.309 + 0.951i)25-s + (2.65 − 2.65i)27-s + (0.951 − 0.690i)31-s + (2.34 − 1.70i)36-s + (1.39 − 0.221i)37-s + (−2.34 + 1.70i)45-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)3-s + (−0.587 − 0.809i)4-s + (0.587 + 0.809i)5-s + 2.90i·9-s + (−0.309 + 1.95i)12-s + (0.309 − 1.95i)15-s + (−0.309 + 0.951i)16-s + (0.309 − 0.951i)20-s + (−0.142 − 0.896i)23-s + (−0.309 + 0.951i)25-s + (2.65 − 2.65i)27-s + (0.951 − 0.690i)31-s + (2.34 − 1.70i)36-s + (1.39 − 0.221i)37-s + (−2.34 + 1.70i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.193 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6622590375\)
\(L(\frac12)\) \(\approx\) \(0.6622590375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T^{2} \)
3 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
7 \( 1 + (0.587 + 0.809i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \)
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

−8.548264370373927672439508855127, −7.72228481400598427328074130355, −6.91459812501717299443120169658, −6.29640212789100357011373689111, −5.88464811011344882433661875142, −5.16343095565612514553427611960, −4.32040181652916367849210740048, −2.56671879989488012307482103848, −1.78099976247861505891920737750, −0.66287643333430241100652299763, 0.984318007098203247144343533775, 2.96020883099893438181964976073, 3.96284119915518683565558892060, 4.55373367306980212711196181606, 5.11054415401084961655654302980, 5.85272334354242789675260216211, 6.53312550415063136259474350049, 7.72958473035764495739419746726, 8.651250077286857973740839436952, 9.398265896036385300817280793081

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