Properties

Label 2-460e2-1.1-c1-0-34
Degree $2$
Conductor $211600$
Sign $-1$
Analytic cond. $1689.63$
Root an. cond. $41.1051$
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 − 2·7-s + 9-s − 2·13-s − 6·17-s − 4·19-s + 4·21-s + 4·27-s + 6·29-s + 4·31-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s − 6·53-s + 8·57-s − 12·59-s − 2·61-s − 2·63-s − 2·67-s + 12·71-s − 2·73-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(211600\)    =    \(2^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1689.63\)
Root analytic conductor: \(41.1051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 211600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.00228418619092, −12.75665341872606, −12.23237309737730, −11.97314986962569, −11.24166099068490, −10.90445323361229, −10.67518142999101, −10.00027913760533, −9.555907532779929, −9.086226047208854, −8.569591601682723, −7.990499620368098, −7.470235081329580, −6.638108540374705, −6.547725124612121, −6.156470769739191, −5.614695357288647, −4.796884634118764, −4.635562562247647, −4.112463655446853, −3.269526651404290, −2.646934085891658, −2.264090339457877, −1.319394610645741, −0.5450275462336395, 0, 0.5450275462336395, 1.319394610645741, 2.264090339457877, 2.646934085891658, 3.269526651404290, 4.112463655446853, 4.635562562247647, 4.796884634118764, 5.614695357288647, 6.156470769739191, 6.547725124612121, 6.638108540374705, 7.470235081329580, 7.990499620368098, 8.569591601682723, 9.086226047208854, 9.555907532779929, 10.00027913760533, 10.67518142999101, 10.90445323361229, 11.24166099068490, 11.97314986962569, 12.23237309737730, 12.75665341872606, 13.00228418619092

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