Properties

Label 2-2450-1.1-c3-0-133
Degree $2$
Conductor $2450$
Sign $-1$
Analytic cond. $144.554$
Root an. cond. $12.0230$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 10·3-s + 4·4-s − 20·6-s + 8·8-s + 73·9-s + 53·11-s − 40·12-s − 25·13-s + 16·16-s − 14·17-s + 146·18-s − 95·19-s + 106·22-s − 23-s − 80·24-s − 50·26-s − 460·27-s − 206·29-s + 108·31-s + 32·32-s − 530·33-s − 28·34-s + 292·36-s + 57·37-s − 190·38-s + 250·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.92·3-s + 1/2·4-s − 1.36·6-s + 0.353·8-s + 2.70·9-s + 1.45·11-s − 0.962·12-s − 0.533·13-s + 1/4·16-s − 0.199·17-s + 1.91·18-s − 1.14·19-s + 1.02·22-s − 0.00906·23-s − 0.680·24-s − 0.377·26-s − 3.27·27-s − 1.31·29-s + 0.625·31-s + 0.176·32-s − 2.79·33-s − 0.141·34-s + 1.35·36-s + 0.253·37-s − 0.811·38-s + 1.02·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(144.554\)
Root analytic conductor: \(12.0230\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 10 T + p^{3} T^{2} \)
11 \( 1 - 53 T + p^{3} T^{2} \)
13 \( 1 + 25 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 5 p T + p^{3} T^{2} \)
23 \( 1 + T + p^{3} T^{2} \)
29 \( 1 + 206 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 - 57 T + p^{3} T^{2} \)
41 \( 1 - 243 T + p^{3} T^{2} \)
43 \( 1 + 434 T + p^{3} T^{2} \)
47 \( 1 - 231 T + p^{3} T^{2} \)
53 \( 1 + 263 T + p^{3} T^{2} \)
59 \( 1 - 24 T + p^{3} T^{2} \)
61 \( 1 - 116 T + p^{3} T^{2} \)
67 \( 1 - 204 T + p^{3} T^{2} \)
71 \( 1 - 484 T + p^{3} T^{2} \)
73 \( 1 - 692 T + p^{3} T^{2} \)
79 \( 1 - 466 T + p^{3} T^{2} \)
83 \( 1 + 228 T + p^{3} T^{2} \)
89 \( 1 + 362 T + p^{3} T^{2} \)
97 \( 1 + 854 T + p^{3} T^{2} \)
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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

−7.907367230250473392318594360826, −6.87607965801577406657530133667, −6.57003229162526716316249900321, −5.86460102007572322975612813057, −5.09920775524285571892109242985, −4.34383220161283780726321582312, −3.78462121035277861232605629933, −2.08425202719541802662812755553, −1.13143278168042238417467294541, 0, 1.13143278168042238417467294541, 2.08425202719541802662812755553, 3.78462121035277861232605629933, 4.34383220161283780726321582312, 5.09920775524285571892109242985, 5.86460102007572322975612813057, 6.57003229162526716316249900321, 6.87607965801577406657530133667, 7.907367230250473392318594360826

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