Properties

Label 2-2450-1.1-c1-0-49
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s − 4·13-s + 16-s + 6·17-s + 18-s − 2·19-s − 2·24-s − 4·26-s + 4·27-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s − 2·38-s + 8·39-s − 6·41-s − 8·43-s − 12·47-s − 2·48-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.408·24-s − 0.784·26-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + 1.28·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s − 0.288·48-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: \(1\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.358573952238112860394001946174, −7.60556812848957435561623382149, −6.73571587686584923478967639363, −6.15190325448345686265108903037, −5.12614454716073682582164065811, −5.04607295657079957887442801881, −3.77049860941673777607586608067, −2.83590121110111537782644859852, −1.54150783577442522625509630661, 0, 1.54150783577442522625509630661, 2.83590121110111537782644859852, 3.77049860941673777607586608067, 5.04607295657079957887442801881, 5.12614454716073682582164065811, 6.15190325448345686265108903037, 6.73571587686584923478967639363, 7.60556812848957435561623382149, 8.358573952238112860394001946174

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