Properties

Label 2-8025-1.1-c1-0-65
Degree $2$
Conductor $8025$
Sign $1$
Analytic cond. $64.0799$
Root an. cond. $8.00499$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.514·2-s − 3-s − 1.73·4-s − 0.514·6-s + 1.81·7-s − 1.92·8-s + 9-s + 1.06·11-s + 1.73·12-s − 1.63·13-s + 0.932·14-s + 2.47·16-s + 6.28·17-s + 0.514·18-s − 7.69·19-s − 1.81·21-s + 0.546·22-s + 6.24·23-s + 1.92·24-s − 0.843·26-s − 27-s − 3.14·28-s − 1.20·29-s + 4.02·31-s + 5.12·32-s − 1.06·33-s + 3.23·34-s + ⋯
L(s)  = 1  + 0.364·2-s − 0.577·3-s − 0.867·4-s − 0.210·6-s + 0.684·7-s − 0.679·8-s + 0.333·9-s + 0.320·11-s + 0.500·12-s − 0.454·13-s + 0.249·14-s + 0.619·16-s + 1.52·17-s + 0.121·18-s − 1.76·19-s − 0.395·21-s + 0.116·22-s + 1.30·23-s + 0.392·24-s − 0.165·26-s − 0.192·27-s − 0.593·28-s − 0.224·29-s + 0.723·31-s + 0.905·32-s − 0.184·33-s + 0.554·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8025\)    =    \(3 \cdot 5^{2} \cdot 107\)
Sign: $1$
Analytic conductor: \(64.0799\)
Root analytic conductor: \(8.00499\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.543885701\)
\(L(\frac12)\) \(\approx\) \(1.543885701\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
107 \( 1 - T \)
good2 \( 1 - 0.514T + 2T^{2} \)
7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 - 1.06T + 11T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 + 7.69T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 + 1.20T + 29T^{2} \)
31 \( 1 - 4.02T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 5.53T + 43T^{2} \)
47 \( 1 + 1.69T + 47T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 + 7.38T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 0.0182T + 71T^{2} \)
73 \( 1 + 2.32T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 - 7.72T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 2.72T + 97T^{2} \)
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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

−7.902427032047932888819656280543, −7.09117473989696794416759394441, −6.26444978133691504382312102836, −5.63857877863339058618477649699, −4.92488314335744388812617618116, −4.50326972642553439010268111776, −3.71128514774569970687111687683, −2.82305294362512303301115505162, −1.59606425910783968269463515152, −0.63364280307848728191314761780, 0.63364280307848728191314761780, 1.59606425910783968269463515152, 2.82305294362512303301115505162, 3.71128514774569970687111687683, 4.50326972642553439010268111776, 4.92488314335744388812617618116, 5.63857877863339058618477649699, 6.26444978133691504382312102836, 7.09117473989696794416759394441, 7.902427032047932888819656280543

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