Properties

Degree $2$
Conductor $317900$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 9-s + 11-s + 4·13-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s + 6·29-s − 8·31-s − 2·33-s + 2·37-s − 8·39-s − 6·41-s − 8·43-s − 6·47-s + 9·49-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s − 4·63-s + 10·67-s + 12·69-s + 12·71-s − 16·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s + 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(317900\)    =    \(2^{2} \cdot 5^{2} \cdot 11 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{317900} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 317900,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.80206643818519, −12.39710737743799, −11.89365269429954, −11.60178527287197, −11.00843535570219, −10.57654373846570, −10.26600596657329, −9.739957229709472, −9.334482213088615, −8.609485278189752, −8.448984103220546, −7.774920496346804, −6.892110854579573, −6.747504393408174, −6.196074168211381, −6.031565412744854, −5.473504133404292, −4.886737073814798, −4.244936269349019, −3.765426944214685, −3.302418177993942, −2.751700500332949, −1.911410435621373, −1.347362911336997, −0.4825516685335618, 0, 0.4825516685335618, 1.347362911336997, 1.911410435621373, 2.751700500332949, 3.302418177993942, 3.765426944214685, 4.244936269349019, 4.886737073814798, 5.473504133404292, 6.031565412744854, 6.196074168211381, 6.747504393408174, 6.892110854579573, 7.774920496346804, 8.448984103220546, 8.609485278189752, 9.334482213088615, 9.739957229709472, 10.26600596657329, 10.57654373846570, 11.00843535570219, 11.60178527287197, 11.89365269429954, 12.39710737743799, 12.80206643818519

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