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Properties

Label 2-217800-1.1-c1-0-103
Degree $2$
Conductor $217800$
Sign $-1$
Analytic cond. $1739.14$
Root an. cond. $41.7030$
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  − 7-s + 3·13-s − 7·17-s − 7·19-s + 6·23-s + 29-s + 2·31-s + 37-s − 6·41-s − 12·43-s + 2·47-s − 6·49-s − 8·59-s + 10·61-s − 4·67-s − 7·71-s + 16·73-s − 4·79-s − 83-s + 2·89-s − 3·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.832·13-s − 1.69·17-s − 1.60·19-s + 1.25·23-s + 0.185·29-s + 0.359·31-s + 0.164·37-s − 0.937·41-s − 1.82·43-s + 0.291·47-s − 6/7·49-s − 1.04·59-s + 1.28·61-s − 0.488·67-s − 0.830·71-s + 1.87·73-s − 0.450·79-s − 0.109·83-s + 0.211·89-s − 0.314·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 217800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 217800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(217800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1739.14\)
Root analytic conductor: \(41.7030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 217800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good7 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.09388572037072, −12.88545941135807, −12.45310095574644, −11.67910250270159, −11.21467166250954, −11.05923775808464, −10.37343242868702, −10.07266605942922, −9.400104220883283, −8.738550623826532, −8.696324198914375, −8.199618556871119, −7.462966670008862, −6.757622489679902, −6.564715607296221, −6.241758019940451, −5.478959488730159, −4.787832951542956, −4.533770579890941, −3.864996579503985, −3.310598312554411, −2.773891185770158, −2.041461967710908, −1.636561371388666, −0.6865439168351810, 0, 0.6865439168351810, 1.636561371388666, 2.041461967710908, 2.773891185770158, 3.310598312554411, 3.864996579503985, 4.533770579890941, 4.787832951542956, 5.478959488730159, 6.241758019940451, 6.564715607296221, 6.757622489679902, 7.462966670008862, 8.199618556871119, 8.696324198914375, 8.738550623826532, 9.400104220883283, 10.07266605942922, 10.37343242868702, 11.05923775808464, 11.21467166250954, 11.67910250270159, 12.45310095574644, 12.88545941135807, 13.09388572037072

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