Properties

Degree $2$
Conductor $2450$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s − 2·11-s + 3·12-s + 16-s − 4·17-s − 6·18-s + 6·19-s + 2·22-s − 3·23-s − 3·24-s + 9·27-s + 9·29-s + 4·31-s − 32-s − 6·33-s + 4·34-s + 6·36-s + 4·37-s − 6·38-s + 7·41-s + 5·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.603·11-s + 0.866·12-s + 1/4·16-s − 0.970·17-s − 1.41·18-s + 1.37·19-s + 0.426·22-s − 0.625·23-s − 0.612·24-s + 1.73·27-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.685·34-s + 36-s + 0.657·37-s − 0.973·38-s + 1.09·41-s + 0.762·43-s − 0.301·44-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: $1$
Motivic weight: \(1\)
Character: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.521089737\)
\(L(\frac12)\) \(\approx\) \(2.521089737\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.965481288250785898829283008574, −8.153276850381094005493949185211, −7.75482899372874136780437024042, −7.02697016707504450398481452546, −6.07072648860756642778838685688, −4.77487246169842344977010988717, −3.87001358016329068702122423338, −2.78588534104432571077405186020, −2.40510021455596195483136420469, −1.08922700890937652804724612626, 1.08922700890937652804724612626, 2.40510021455596195483136420469, 2.78588534104432571077405186020, 3.87001358016329068702122423338, 4.77487246169842344977010988717, 6.07072648860756642778838685688, 7.02697016707504450398481452546, 7.75482899372874136780437024042, 8.153276850381094005493949185211, 8.965481288250785898829283008574

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