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Properties

Label 2-95e2-1.1-c1-0-471
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
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 − 2·4-s + 4·7-s + 9-s + 3·11-s − 4·12-s − 2·13-s + 4·16-s − 6·17-s + 8·21-s − 4·27-s − 8·28-s − 3·29-s − 7·31-s + 6·33-s − 2·36-s − 8·37-s − 4·39-s − 6·41-s + 4·43-s − 6·44-s − 6·47-s + 8·48-s + 9·49-s − 12·51-s + 4·52-s + 6·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.51·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s − 0.554·13-s + 16-s − 1.45·17-s + 1.74·21-s − 0.769·27-s − 1.51·28-s − 0.557·29-s − 1.25·31-s + 1.04·33-s − 1/3·36-s − 1.31·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.904·44-s − 0.875·47-s + 1.15·48-s + 9/7·49-s − 1.68·51-s + 0.554·52-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.54077871130760231857953639957, −7.02670731199975696065066202804, −5.86857745605201058059579690124, −5.10667495915163109837412560658, −4.48119941130551282907573401981, −3.93584614361416858298673264501, −3.15451099160830095639023268055, −2.02950889034593554805831885097, −1.55378852660460568540896216770, 0, 1.55378852660460568540896216770, 2.02950889034593554805831885097, 3.15451099160830095639023268055, 3.93584614361416858298673264501, 4.48119941130551282907573401981, 5.10667495915163109837412560658, 5.86857745605201058059579690124, 7.02670731199975696065066202804, 7.54077871130760231857953639957

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