Properties

Label 2-1925-1.1-c1-0-44
Degree $2$
Conductor $1925$
Sign $1$
Analytic cond. $15.3712$
Root an. cond. $3.92061$
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  − 2.46·2-s + 2.19·3-s + 4.06·4-s − 5.41·6-s + 7-s − 5.09·8-s + 1.83·9-s + 11-s + 8.94·12-s + 6.39·13-s − 2.46·14-s + 4.41·16-s + 5.64·17-s − 4.50·18-s + 1.70·19-s + 2.19·21-s − 2.46·22-s + 3.34·23-s − 11.1·24-s − 15.7·26-s − 2.57·27-s + 4.06·28-s + 0.132·29-s − 2.25·31-s − 0.678·32-s + 2.19·33-s − 13.9·34-s + ⋯
L(s)  = 1  − 1.74·2-s + 1.26·3-s + 2.03·4-s − 2.21·6-s + 0.377·7-s − 1.80·8-s + 0.610·9-s + 0.301·11-s + 2.58·12-s + 1.77·13-s − 0.658·14-s + 1.10·16-s + 1.36·17-s − 1.06·18-s + 0.390·19-s + 0.479·21-s − 0.525·22-s + 0.698·23-s − 2.28·24-s − 3.09·26-s − 0.494·27-s + 0.768·28-s + 0.0245·29-s − 0.405·31-s − 0.119·32-s + 0.382·33-s − 2.38·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(15.3712\)
Root analytic conductor: \(3.92061\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.478754711\)
\(L(\frac12)\) \(\approx\) \(1.478754711\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 - T \)
good2 \( 1 + 2.46T + 2T^{2} \)
3 \( 1 - 2.19T + 3T^{2} \)
13 \( 1 - 6.39T + 13T^{2} \)
17 \( 1 - 5.64T + 17T^{2} \)
19 \( 1 - 1.70T + 19T^{2} \)
23 \( 1 - 3.34T + 23T^{2} \)
29 \( 1 - 0.132T + 29T^{2} \)
31 \( 1 + 2.25T + 31T^{2} \)
37 \( 1 + 2.84T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 4.05T + 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 + 4.64T + 53T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 3.99T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + 4.71T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−9.098895620763154817938819604202, −8.312669215541426774931431768627, −8.174470112733175948734522368529, −7.26527041115544284468592524792, −6.46573174181135729817655893905, −5.37148761584817229725905173107, −3.71650257433698672948869402774, −3.09129739007887637080240417084, −1.84529325542242915870728501572, −1.10321816457297096579791375966, 1.10321816457297096579791375966, 1.84529325542242915870728501572, 3.09129739007887637080240417084, 3.71650257433698672948869402774, 5.37148761584817229725905173107, 6.46573174181135729817655893905, 7.26527041115544284468592524792, 8.174470112733175948734522368529, 8.312669215541426774931431768627, 9.098895620763154817938819604202

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