Properties

Label 2-35e2-1.1-c1-0-49
Degree $2$
Conductor $1225$
Sign $-1$
Analytic cond. $9.78167$
Root an. cond. $3.12756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 0.414·3-s − 0.585·6-s − 2.82·8-s − 2.82·9-s − 0.171·11-s + 4.41·13-s − 4.00·16-s + 3.24·17-s − 4.00·18-s − 6·19-s − 0.242·22-s − 7.41·23-s + 1.17·24-s + 6.24·26-s + 2.41·27-s − 8.65·29-s − 10.2·31-s + 0.0710·33-s + 4.58·34-s − 2.24·37-s − 8.48·38-s − 1.82·39-s + 6.24·41-s − 2·43-s + ⋯
L(s)  = 1  + 1.00·2-s − 0.239·3-s − 0.239·6-s − 0.999·8-s − 0.942·9-s − 0.0517·11-s + 1.22·13-s − 1.00·16-s + 0.786·17-s − 0.942·18-s − 1.37·19-s − 0.0517·22-s − 1.54·23-s + 0.239·24-s + 1.22·26-s + 0.464·27-s − 1.60·29-s − 1.83·31-s + 0.0123·33-s + 0.786·34-s − 0.368·37-s − 1.37·38-s − 0.292·39-s + 0.974·41-s − 0.304·43-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: no
Arithmetic: yes
Character: $\chi_{1225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1225,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good2 \( 1 - 1.41T + 2T^{2} \)
3 \( 1 + 0.414T + 3T^{2} \)
11 \( 1 + 0.171T + 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 - 3.24T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 2.24T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 7.24T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 8.24T + 67T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 1.48T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−9.162712528183148625502420888011, −8.582824988376858553763649652553, −7.69244691157790295330949432186, −6.29135173236736734513947731879, −5.90363246031709174367358180034, −5.14520379377303214158004984831, −3.93853724455388037122141165881, −3.45465562630705335778475981188, −2.05010198598644858450103290804, 0, 2.05010198598644858450103290804, 3.45465562630705335778475981188, 3.93853724455388037122141165881, 5.14520379377303214158004984831, 5.90363246031709174367358180034, 6.29135173236736734513947731879, 7.69244691157790295330949432186, 8.582824988376858553763649652553, 9.162712528183148625502420888011

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