Properties

Label 2-35e2-1.1-c1-0-12
Degree $2$
Conductor $1225$
Sign $1$
Analytic cond. $9.78167$
Root an. cond. $3.12756$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s + 4·16-s + 3·17-s − 2·19-s + 6·23-s − 5·27-s + 3·29-s + 4·31-s − 3·33-s + 4·36-s − 2·37-s + 5·39-s + 12·41-s + 10·43-s + 6·44-s + 9·47-s + 4·48-s + 3·51-s − 10·52-s − 12·53-s − 2·57-s − 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.328·37-s + 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.904·44-s + 1.31·47-s + 0.577·48-s + 0.420·51-s − 1.38·52-s − 1.64·53-s − 0.264·57-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.78167\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490882041\)
\(L(\frac12)\) \(\approx\) \(1.490882041\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−9.408054549667512195696276237510, −8.942402513263511435033585555615, −8.173579333842050044011512625481, −7.65396811735679492201158087709, −6.18506570501734980145358588299, −5.49929554576320674834936734485, −4.50035894020182177306684850522, −3.50518882381413089746428044840, −2.67848601395212920300767663698, −0.913243850144851748702778420825, 0.913243850144851748702778420825, 2.67848601395212920300767663698, 3.50518882381413089746428044840, 4.50035894020182177306684850522, 5.49929554576320674834936734485, 6.18506570501734980145358588299, 7.65396811735679492201158087709, 8.173579333842050044011512625481, 8.942402513263511435033585555615, 9.408054549667512195696276237510

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