Properties

Label 2-4025-1.1-c1-0-132
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  − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s − 5·11-s + 2·12-s − 3·13-s − 2·14-s − 4·16-s + 5·17-s + 4·18-s + 21-s + 10·22-s + 23-s + 6·26-s − 5·27-s + 2·28-s + 3·29-s + 6·31-s + 8·32-s − 5·33-s − 10·34-s − 4·36-s + 4·37-s − 3·39-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.577·12-s − 0.832·13-s − 0.534·14-s − 16-s + 1.21·17-s + 0.942·18-s + 0.218·21-s + 2.13·22-s + 0.208·23-s + 1.17·26-s − 0.962·27-s + 0.377·28-s + 0.557·29-s + 1.07·31-s + 1.41·32-s − 0.870·33-s − 1.71·34-s − 2/3·36-s + 0.657·37-s − 0.480·39-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: $\chi_{4025} (1, \cdot )$
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 \( 1 + p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.176384922727208447230330582813, −7.63807370154589821455136394048, −7.16364310068840225553097410474, −5.87569790442897986888279234516, −5.18979928366786942928980270321, −4.26683137334565487932389416652, −2.78772473421367586501771496107, −2.54579250009981993592100560124, −1.20742621489710710952356534454, 0, 1.20742621489710710952356534454, 2.54579250009981993592100560124, 2.78772473421367586501771496107, 4.26683137334565487932389416652, 5.18979928366786942928980270321, 5.87569790442897986888279234516, 7.16364310068840225553097410474, 7.63807370154589821455136394048, 8.176384922727208447230330582813

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