Properties

Label 2-9800-1.1-c1-0-50
Degree $2$
Conductor $9800$
Sign $1$
Analytic cond. $78.2533$
Root an. cond. $8.84609$
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  + 0.402·3-s − 2.83·9-s − 5.46·11-s + 4.97·13-s + 6.78·17-s + 0.859·19-s + 5.40·23-s − 2.34·27-s − 9.29·29-s − 6.99·31-s − 2.19·33-s + 2.76·37-s + 2·39-s + 9.53·41-s − 4.76·43-s + 11.1·47-s + 2.73·51-s − 1.89·53-s + 0.345·57-s − 11.2·59-s + 5.50·61-s − 3.86·67-s + 2.17·69-s − 10.4·71-s + 0.173·73-s + 5.13·79-s + 7.56·81-s + ⋯
L(s)  = 1  + 0.232·3-s − 0.946·9-s − 1.64·11-s + 1.37·13-s + 1.64·17-s + 0.197·19-s + 1.12·23-s − 0.452·27-s − 1.72·29-s − 1.25·31-s − 0.382·33-s + 0.453·37-s + 0.320·39-s + 1.48·41-s − 0.725·43-s + 1.62·47-s + 0.382·51-s − 0.259·53-s + 0.0458·57-s − 1.45·59-s + 0.704·61-s − 0.471·67-s + 0.261·69-s − 1.23·71-s + 0.0202·73-s + 0.577·79-s + 0.841·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9800\)    =    \(2^{3} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(78.2533\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.841199615\)
\(L(\frac12)\) \(\approx\) \(1.841199615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 0.402T + 3T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 - 6.78T + 17T^{2} \)
19 \( 1 - 0.859T + 19T^{2} \)
23 \( 1 - 5.40T + 23T^{2} \)
29 \( 1 + 9.29T + 29T^{2} \)
31 \( 1 + 6.99T + 31T^{2} \)
37 \( 1 - 2.76T + 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 1.89T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
67 \( 1 + 3.86T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 0.173T + 73T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 - 6.43T + 83T^{2} \)
89 \( 1 - 6.51T + 89T^{2} \)
97 \( 1 + 8.94T + 97T^{2} \)
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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

−7.74292075299848921889904057903, −7.24958979622363551079442245140, −6.03767301035042976003311903142, −5.59480946596119547067620530836, −5.22289933940531129881905591150, −4.02287743962002597718795157426, −3.25940830602218741463707463944, −2.81375400268842282679264865380, −1.74567696009728654517620965700, −0.63257193507613398915665298263, 0.63257193507613398915665298263, 1.74567696009728654517620965700, 2.81375400268842282679264865380, 3.25940830602218741463707463944, 4.02287743962002597718795157426, 5.22289933940531129881905591150, 5.59480946596119547067620530836, 6.03767301035042976003311903142, 7.24958979622363551079442245140, 7.74292075299848921889904057903

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