Properties

Label 2-35e2-1.1-c1-0-20
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 − 2.41·3-s + 3.41·6-s + 2.82·8-s + 2.82·9-s − 5.82·11-s − 1.58·13-s − 4.00·16-s + 5.24·17-s − 4·18-s + 6·19-s + 8.24·22-s − 4.58·23-s − 6.82·24-s + 2.24·26-s + 0.414·27-s + 2.65·29-s + 1.75·31-s + 14.0·33-s − 7.41·34-s + 6.24·37-s − 8.48·38-s + 3.82·39-s + 2.24·41-s − 2·43-s + ⋯
L(s)  = 1  − 1.00·2-s − 1.39·3-s + 1.39·6-s + 0.999·8-s + 0.942·9-s − 1.75·11-s − 0.439·13-s − 1.00·16-s + 1.27·17-s − 0.942·18-s + 1.37·19-s + 1.75·22-s − 0.956·23-s − 1.39·24-s + 0.439·26-s + 0.0797·27-s + 0.493·29-s + 0.315·31-s + 2.44·33-s − 1.27·34-s + 1.02·37-s − 1.37·38-s + 0.613·39-s + 0.350·41-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 \( 1 + 2.41T + 3T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 4.58T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 - 6.24T + 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 1.24T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + 0.242T + 67T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + 4.75T + 97T^{2} \)
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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

−9.727562904682335687954883051916, −8.351562204940317397312430777185, −7.74466456519532638811131211242, −7.07874004748954413403443473732, −5.71512687113582828835604201501, −5.32438407152611259801184244192, −4.40556925606388490498667873035, −2.78840361055997743144197075966, −1.12557278961490605786276875128, 0, 1.12557278961490605786276875128, 2.78840361055997743144197075966, 4.40556925606388490498667873035, 5.32438407152611259801184244192, 5.71512687113582828835604201501, 7.07874004748954413403443473732, 7.74466456519532638811131211242, 8.351562204940317397312430777185, 9.727562904682335687954883051916

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