Properties

Label 2-45e2-1.1-c1-0-25
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  − 0.792·2-s − 1.37·4-s + 3.46·7-s + 2.67·8-s + 4.37·11-s + 5.84·13-s − 2.74·14-s + 0.627·16-s − 0.792·17-s − 0.372·19-s − 3.46·22-s − 1.58·23-s − 4.62·26-s − 4.75·28-s + 5.74·29-s + 6.37·31-s − 5.84·32-s + 0.627·34-s − 2.37·37-s + 0.294·38-s − 4.37·41-s − 3.46·43-s − 6·44-s + 1.25·46-s − 1.87·47-s + 4.99·49-s − 8.01·52-s + ⋯
L(s)  = 1  − 0.560·2-s − 0.686·4-s + 1.30·7-s + 0.944·8-s + 1.31·11-s + 1.61·13-s − 0.733·14-s + 0.156·16-s − 0.192·17-s − 0.0854·19-s − 0.738·22-s − 0.330·23-s − 0.907·26-s − 0.898·28-s + 1.06·29-s + 1.14·31-s − 1.03·32-s + 0.107·34-s − 0.390·37-s + 0.0478·38-s − 0.682·41-s − 0.528·43-s − 0.904·44-s + 0.185·46-s − 0.274·47-s + 0.714·49-s − 1.11·52-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

\(L(1)\) \(\approx\) \(1.567091340\)
\(L(\frac12)\) \(\approx\) \(1.567091340\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.792T + 2T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 - 5.84T + 13T^{2} \)
17 \( 1 + 0.792T + 17T^{2} \)
19 \( 1 + 0.372T + 19T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 - 5.74T + 29T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 + 2.37T + 37T^{2} \)
41 \( 1 + 4.37T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + 1.87T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 1.62T + 59T^{2} \)
61 \( 1 - 9.37T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + 1.28T + 97T^{2} \)
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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

−8.894566576227192249143690164412, −8.449090817621440080766190057018, −7.997141604321380270475986890317, −6.82587092737947368362695415706, −6.06572854375836656557161130859, −4.92871129816834904903256380569, −4.31692788287165865990642589343, −3.48137038560412142421624022065, −1.71162841224541497355043549549, −1.04717586211223452855631539167, 1.04717586211223452855631539167, 1.71162841224541497355043549549, 3.48137038560412142421624022065, 4.31692788287165865990642589343, 4.92871129816834904903256380569, 6.06572854375836656557161130859, 6.82587092737947368362695415706, 7.997141604321380270475986890317, 8.449090817621440080766190057018, 8.894566576227192249143690164412

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