Properties

Label 2-2450-1.1-c1-0-25
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 3·9-s + 3·11-s + 5·13-s + 16-s + 2·17-s − 3·18-s + 5·19-s + 3·22-s − 7·23-s + 5·26-s − 4·29-s + 2·31-s + 32-s + 2·34-s − 3·36-s + 37-s + 5·38-s − 3·41-s + 2·43-s + 3·44-s − 7·46-s + 7·47-s + 5·52-s + 9·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 0.904·11-s + 1.38·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.14·19-s + 0.639·22-s − 1.45·23-s + 0.980·26-s − 0.742·29-s + 0.359·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.164·37-s + 0.811·38-s − 0.468·41-s + 0.304·43-s + 0.452·44-s − 1.03·46-s + 1.02·47-s + 0.693·52-s + 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.053496384\)
\(L(\frac12)\) \(\approx\) \(3.053496384\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−8.884378773706066469879765673919, −8.145906943090622224012208291388, −7.36829428213710417166599671527, −6.28106962023091567989287023729, −5.91269983513673888182193368138, −5.10383000747442847131603475924, −3.84131051985661067682741046216, −3.51224935072617371017041219114, −2.29244831296924493589688319369, −1.06615783429189792116447319441, 1.06615783429189792116447319441, 2.29244831296924493589688319369, 3.51224935072617371017041219114, 3.84131051985661067682741046216, 5.10383000747442847131603475924, 5.91269983513673888182193368138, 6.28106962023091567989287023729, 7.36829428213710417166599671527, 8.145906943090622224012208291388, 8.884378773706066469879765673919

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