Properties

Label 2-257600-1.1-c1-0-122
Degree $2$
Conductor $257600$
Sign $-1$
Analytic cond. $2056.94$
Root an. cond. $45.3535$
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  − 2·3-s − 7-s + 9-s + 4·11-s + 6·13-s + 2·21-s + 23-s + 4·27-s − 2·29-s − 2·31-s − 8·33-s + 2·37-s − 12·39-s − 6·41-s − 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s − 12·61-s − 63-s + 4·67-s − 2·69-s + 2·73-s − 4·77-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 0.371·29-s − 0.359·31-s − 1.39·33-s + 0.328·37-s − 1.92·39-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s − 1.53·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 0.234·73-s − 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(257600\)    =    \(2^{6} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(2056.94\)
Root analytic conductor: \(45.3535\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 257600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.12544233644331, −12.46375799375848, −11.99760576115676, −11.65435905314851, −11.25500334317636, −10.89284960360497, −10.38570607176451, −9.938128790551428, −9.303250872920562, −8.889187258683773, −8.481642369206326, −7.982636271502178, −7.153727281792670, −6.648439109732981, −6.535878441934546, −5.896237698914970, −5.518534798474819, −5.076985511856857, −4.259488908761391, −3.881003163357587, −3.445485533219894, −2.761650265502228, −1.885108344993052, −1.250842509597153, −0.8242180484062006, 0, 0.8242180484062006, 1.250842509597153, 1.885108344993052, 2.761650265502228, 3.445485533219894, 3.881003163357587, 4.259488908761391, 5.076985511856857, 5.518534798474819, 5.896237698914970, 6.535878441934546, 6.648439109732981, 7.153727281792670, 7.982636271502178, 8.481642369206326, 8.889187258683773, 9.303250872920562, 9.938128790551428, 10.38570607176451, 10.89284960360497, 11.25500334317636, 11.65435905314851, 11.99760576115676, 12.46375799375848, 13.12544233644331

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