↑

Properties

Label 2-3875-3875.1239-c0-0-0
Degree $2$
Conductor $3875$
Sign $-0.988 + 0.150i$
Analytic cond. $1.93387$
Root an. cond. $1.39063$
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  + (1.34 + 1.43i)2-s + (−0.179 + 2.85i)4-s + (−0.876 + 0.481i)5-s + (1.60 + 0.521i)7-s + (−2.81 + 2.32i)8-s + (0.425 − 0.904i)9-s + (−1.86 − 0.607i)10-s + (1.41 + 3.00i)14-s + (−4.28 − 0.541i)16-s + (1.86 − 0.607i)18-s + (−1.03 + 1.63i)19-s + (−1.21 − 2.58i)20-s + (0.535 − 0.844i)25-s + (−1.77 + 4.48i)28-s + (−0.0627 − 0.998i)31-s + (−2.83 − 3.90i)32-s + ⋯
L(s)  = 1  + (1.34 + 1.43i)2-s + (−0.179 + 2.85i)4-s + (−0.876 + 0.481i)5-s + (1.60 + 0.521i)7-s + (−2.81 + 2.32i)8-s + (0.425 − 0.904i)9-s + (−1.86 − 0.607i)10-s + (1.41 + 3.00i)14-s + (−4.28 − 0.541i)16-s + (1.86 − 0.607i)18-s + (−1.03 + 1.63i)19-s + (−1.21 − 2.58i)20-s + (0.535 − 0.844i)25-s + (−1.77 + 4.48i)28-s + (−0.0627 − 0.998i)31-s + (−2.83 − 3.90i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3875\)    =    \(5^{3} \cdot 31\)
Sign: $-0.988 + 0.150i$
Analytic conductor: \(1.93387\)
Root analytic conductor: \(1.39063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3875} (1239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3875,\ (\ :0),\ -0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.486276066\)
\(L(\frac12)\) \(\approx\) \(2.486276066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.876 - 0.481i)T \)
31 \( 1 + (0.0627 + 0.998i)T \)
good2 \( 1 + (-1.34 - 1.43i)T + (-0.0627 + 0.998i)T^{2} \)
3 \( 1 + (-0.425 + 0.904i)T^{2} \)
7 \( 1 + (-1.60 - 0.521i)T + (0.809 + 0.587i)T^{2} \)
11 \( 1 + (-0.0627 + 0.998i)T^{2} \)
13 \( 1 + (-0.637 - 0.770i)T^{2} \)
17 \( 1 + (-0.992 + 0.125i)T^{2} \)
19 \( 1 + (1.03 - 1.63i)T + (-0.425 - 0.904i)T^{2} \)
23 \( 1 + (-0.929 - 0.368i)T^{2} \)
29 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.968 + 0.248i)T^{2} \)
41 \( 1 + (-0.238 - 1.25i)T + (-0.929 + 0.368i)T^{2} \)
43 \( 1 + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.193 + 0.159i)T + (0.187 + 0.982i)T^{2} \)
53 \( 1 + (0.876 - 0.481i)T^{2} \)
59 \( 1 + (-0.939 + 0.516i)T + (0.535 - 0.844i)T^{2} \)
61 \( 1 + (0.929 + 0.368i)T^{2} \)
67 \( 1 + (0.271 + 0.684i)T + (-0.728 + 0.684i)T^{2} \)
71 \( 1 + (-1.27 + 1.54i)T + (-0.187 - 0.982i)T^{2} \)
73 \( 1 + (0.535 + 0.844i)T^{2} \)
79 \( 1 + (0.425 - 0.904i)T^{2} \)
83 \( 1 + (-0.425 - 0.904i)T^{2} \)
89 \( 1 + (-0.535 - 0.844i)T^{2} \)
97 \( 1 + (0.354 - 0.895i)T + (-0.728 - 0.684i)T^{2} \)
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

−8.403391262477600468421558154239, −8.056312520889459924009823972727, −7.58788819336784740379810099859, −6.65598403512765556279633418174, −6.14375430039270363959479157008, −5.33043635594612615075649360132, −4.46329729394524195368132041251, −4.05201016532425537693086759167, −3.27105679676608279172155665135, −2.08183410868645975801672654606, 0.945719357545956358700456213308, 1.85741812528107969937307001895, 2.65789601149720202809089500401, 3.93866892547556942565119219842, 4.33353067751008975851516622685, 5.01359317480739096507197709052, 5.31589306492190687583094597191, 6.74941506947828322236106894094, 7.43214389830877517910596744455, 8.470515489036329461753938830985

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