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Properties

Label 2-262080-1.1-c1-0-122
Degree $2$
Conductor $262080$
Sign $-1$
Analytic cond. $2092.71$
Root an. cond. $45.7462$
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  − 5-s − 7-s − 4·11-s − 13-s − 2·19-s + 2·23-s + 25-s − 2·29-s + 35-s − 2·37-s − 4·43-s + 49-s + 4·53-s + 4·55-s + 6·59-s − 6·61-s + 65-s + 4·67-s − 6·71-s − 14·73-s + 4·77-s + 8·79-s + 12·83-s + 91-s + 2·95-s − 18·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.277·13-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.169·35-s − 0.328·37-s − 0.609·43-s + 1/7·49-s + 0.549·53-s + 0.539·55-s + 0.781·59-s − 0.768·61-s + 0.124·65-s + 0.488·67-s − 0.712·71-s − 1.63·73-s + 0.455·77-s + 0.900·79-s + 1.31·83-s + 0.104·91-s + 0.205·95-s − 1.82·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(262080\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2092.71\)
Root analytic conductor: \(45.7462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 262080,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.98044564743419, −12.63638110698195, −12.18981817413836, −11.59359186016184, −11.28083940981938, −10.60308380598159, −10.33908101938283, −9.957896677833803, −9.214119493342344, −8.920059325948615, −8.309495332025123, −7.889069655000736, −7.432077942629763, −6.984508755771942, −6.458568524024040, −5.897923390692622, −5.296147483830498, −4.975193442933471, −4.340880483744958, −3.803132625096646, −3.213732489203540, −2.720747964437380, −2.199247270479987, −1.479417600046911, −0.5896939717986087, 0, 0.5896939717986087, 1.479417600046911, 2.199247270479987, 2.720747964437380, 3.213732489203540, 3.803132625096646, 4.340880483744958, 4.975193442933471, 5.296147483830498, 5.897923390692622, 6.458568524024040, 6.984508755771942, 7.432077942629763, 7.889069655000736, 8.309495332025123, 8.920059325948615, 9.214119493342344, 9.957896677833803, 10.33908101938283, 10.60308380598159, 11.28083940981938, 11.59359186016184, 12.18981817413836, 12.63638110698195, 12.98044564743419

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