Properties

Label 2-2450-1.1-c1-0-13
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
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-s − 1.41·3-s + 4-s − 1.41·6-s + 8-s − 0.999·9-s − 1.41·12-s − 4.24·13-s + 16-s + 5.65·17-s − 0.999·18-s − 4.24·19-s + 6·23-s − 1.41·24-s − 4.24·26-s + 5.65·27-s + 6·29-s − 8.48·31-s + 32-s + 5.65·34-s − 0.999·36-s + 6·37-s − 4.24·38-s + 6·39-s + 8.48·41-s + 12·43-s + 6·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.816·3-s + 0.5·4-s − 0.577·6-s + 0.353·8-s − 0.333·9-s − 0.408·12-s − 1.17·13-s + 0.250·16-s + 1.37·17-s − 0.235·18-s − 0.973·19-s + 1.25·23-s − 0.288·24-s − 0.832·26-s + 1.08·27-s + 1.11·29-s − 1.52·31-s + 0.176·32-s + 0.970·34-s − 0.166·36-s + 0.986·37-s − 0.688·38-s + 0.960·39-s + 1.32·41-s + 1.82·43-s + 0.884·46-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.895473545\)
\(L(\frac12)\) \(\approx\) \(1.895473545\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.019670505779790672816517838744, −7.925379766150347932541290058236, −7.28339236859981557561454961886, −6.40888460698946926827754316453, −5.71943854513071216602311788206, −5.07021925012516361208978147584, −4.36232685110883718481838660303, −3.18954180703198252545995699763, −2.36434866636592381548694087464, −0.816655776271033166365496871235, 0.816655776271033166365496871235, 2.36434866636592381548694087464, 3.18954180703198252545995699763, 4.36232685110883718481838660303, 5.07021925012516361208978147584, 5.71943854513071216602311788206, 6.40888460698946926827754316453, 7.28339236859981557561454961886, 7.925379766150347932541290058236, 9.019670505779790672816517838744

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