Properties

Label 2-45e2-1.1-c1-0-4
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  − 2.52·2-s + 4.37·4-s − 3.46·7-s − 5.98·8-s − 1.37·11-s + 4.10·13-s + 8.74·14-s + 6.37·16-s − 2.52·17-s + 5.37·19-s + 3.46·22-s − 5.04·23-s − 10.3·26-s − 15.1·28-s − 5.74·29-s + 0.627·31-s − 4.10·32-s + 6.37·34-s − 7.57·37-s − 13.5·38-s + 1.37·41-s + 3.46·43-s − 6.00·44-s + 12.7·46-s + 8.51·47-s + 4.99·49-s + 17.9·52-s + ⋯
L(s)  = 1  − 1.78·2-s + 2.18·4-s − 1.30·7-s − 2.11·8-s − 0.413·11-s + 1.13·13-s + 2.33·14-s + 1.59·16-s − 0.612·17-s + 1.23·19-s + 0.738·22-s − 1.05·23-s − 2.03·26-s − 2.86·28-s − 1.06·29-s + 0.112·31-s − 0.726·32-s + 1.09·34-s − 1.24·37-s − 2.19·38-s + 0.214·41-s + 0.528·43-s − 0.904·44-s + 1.87·46-s + 1.24·47-s + 0.714·49-s + 2.49·52-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

\(L(1)\) \(\approx\) \(0.4867011115\)
\(L(\frac12)\) \(\approx\) \(0.4867011115\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.52T + 2T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 - 5.37T + 19T^{2} \)
23 \( 1 + 5.04T + 23T^{2} \)
29 \( 1 + 5.74T + 29T^{2} \)
31 \( 1 - 0.627T + 31T^{2} \)
37 \( 1 + 7.57T + 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 - 5.34T + 53T^{2} \)
59 \( 1 + 7.37T + 59T^{2} \)
61 \( 1 - 3.62T + 61T^{2} \)
67 \( 1 - 8.21T + 67T^{2} \)
71 \( 1 - 4.11T + 71T^{2} \)
73 \( 1 + 7.57T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 5.34T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−9.147568412444890999870999282344, −8.584837041200166531428762971324, −7.68508806652542111018777731528, −7.06006620711352042887209849249, −6.27246707747713533034509224587, −5.60069385889621025499205089528, −3.87060206227427494683870901040, −2.97628184686321508236170717143, −1.89093724535037867983692172051, −0.58983640741714363049922167969, 0.58983640741714363049922167969, 1.89093724535037867983692172051, 2.97628184686321508236170717143, 3.87060206227427494683870901040, 5.60069385889621025499205089528, 6.27246707747713533034509224587, 7.06006620711352042887209849249, 7.68508806652542111018777731528, 8.584837041200166531428762971324, 9.147568412444890999870999282344

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