Properties

Label 2-3870-1.1-c1-0-63
Degree $2$
Conductor $3870$
Sign $-1$
Analytic cond. $30.9021$
Root an. cond. $5.55896$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 4·13-s + 16-s + 4·19-s − 20-s − 8·23-s + 25-s − 4·26-s − 2·29-s − 4·31-s + 32-s − 2·37-s + 4·38-s − 40-s − 10·41-s − 43-s − 8·46-s + 12·47-s − 7·49-s + 50-s − 4·52-s − 2·53-s − 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.10·13-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s − 0.152·43-s − 1.17·46-s + 1.75·47-s − 49-s + 0.141·50-s − 0.554·52-s − 0.274·53-s − 0.262·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3870\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(30.9021\)
Root analytic conductor: \(5.55896\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3870,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.88121062094934296124121687182, −7.36617316577632833814789172233, −6.66752208121222356980996238882, −5.69081600221898085162106772500, −5.12557733247462741530059752750, −4.24718414486006713061233724267, −3.55105542891525590393894273039, −2.63672832248773395308304521780, −1.65432668113079297419751568145, 0, 1.65432668113079297419751568145, 2.63672832248773395308304521780, 3.55105542891525590393894273039, 4.24718414486006713061233724267, 5.12557733247462741530059752750, 5.69081600221898085162106772500, 6.66752208121222356980996238882, 7.36617316577632833814789172233, 7.88121062094934296124121687182

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