Properties

Label 2-4025-1.1-c1-0-145
Degree $2$
Conductor $4025$
Sign $-1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
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  + 3-s − 2·4-s − 7-s − 2·9-s − 11-s − 2·12-s + 13-s + 4·16-s + 17-s + 2·19-s − 21-s + 23-s − 5·27-s + 2·28-s + 7·29-s + 4·31-s − 33-s + 4·36-s + 8·37-s + 39-s − 6·41-s − 8·43-s + 2·44-s − 7·47-s + 4·48-s + 49-s + 51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s + 0.277·13-s + 16-s + 0.242·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.377·28-s + 1.29·29-s + 0.718·31-s − 0.174·33-s + 2/3·36-s + 1.31·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.301·44-s − 1.02·47-s + 0.577·48-s + 1/7·49-s + 0.140·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.249051193477960890115594197464, −7.62779720098297255017666351116, −6.53454340706693236827806403764, −5.79992371633617361455709349591, −5.00396804939761087566103974500, −4.26334706188802962808282230504, −3.24057586493752760359257557209, −2.82943919034522223313588838065, −1.31252271658192077335908136807, 0, 1.31252271658192077335908136807, 2.82943919034522223313588838065, 3.24057586493752760359257557209, 4.26334706188802962808282230504, 5.00396804939761087566103974500, 5.79992371633617361455709349591, 6.53454340706693236827806403764, 7.62779720098297255017666351116, 8.249051193477960890115594197464

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