Properties

Label 2-45e2-1.1-c1-0-14
Degree $2$
Conductor $2025$
Sign $1$
Analytic cond. $16.1697$
Root an. cond. $4.02115$
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  − 0.732·2-s − 1.46·4-s + 4.73·7-s + 2.53·8-s − 5.73·11-s − 1.46·13-s − 3.46·14-s + 1.07·16-s + 2.73·17-s + 4.46·19-s + 4.19·22-s + 3.46·23-s + 1.07·26-s − 6.92·28-s + 3.19·29-s − 3·31-s − 5.85·32-s − 2·34-s + 2.73·37-s − 3.26·38-s − 7.19·41-s − 0.196·43-s + 8.39·44-s − 2.53·46-s + 8.73·47-s + 15.3·49-s + 2.14·52-s + ⋯
L(s)  = 1  − 0.517·2-s − 0.732·4-s + 1.78·7-s + 0.896·8-s − 1.72·11-s − 0.406·13-s − 0.925·14-s + 0.267·16-s + 0.662·17-s + 1.02·19-s + 0.894·22-s + 0.722·23-s + 0.210·26-s − 1.30·28-s + 0.593·29-s − 0.538·31-s − 1.03·32-s − 0.342·34-s + 0.449·37-s − 0.530·38-s − 1.12·41-s − 0.0299·43-s + 1.26·44-s − 0.373·46-s + 1.27·47-s + 2.19·49-s + 0.297·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(16.1697\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248225859\)
\(L(\frac12)\) \(\approx\) \(1.248225859\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 0.732T + 2T^{2} \)
7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 + 5.73T + 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 - 2.73T + 17T^{2} \)
19 \( 1 - 4.46T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 2.73T + 37T^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 + 0.196T + 43T^{2} \)
47 \( 1 - 8.73T + 47T^{2} \)
53 \( 1 + 6.73T + 53T^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 - 9.66T + 97T^{2} \)
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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

−9.040314410663308323679746836517, −8.246885561424539140351888020424, −7.76192416674552620936952223598, −7.31985422575701095016963349574, −5.59136789897443639248731596957, −5.05046732407156589214777319240, −4.59200519527533780818264604353, −3.21245120689521848055503110768, −1.98245530703763601898170584253, −0.839772209097735318779632296013, 0.839772209097735318779632296013, 1.98245530703763601898170584253, 3.21245120689521848055503110768, 4.59200519527533780818264604353, 5.05046732407156589214777319240, 5.59136789897443639248731596957, 7.31985422575701095016963349574, 7.76192416674552620936952223598, 8.246885561424539140351888020424, 9.040314410663308323679746836517

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