Properties

Label 2-9800-1.1-c1-0-13
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  − 1.56·3-s − 0.561·9-s − 6.12·11-s + 2·13-s + 1.56·17-s − 3.56·19-s − 1.43·23-s + 5.56·27-s + 3.43·29-s + 9.12·31-s + 9.56·33-s − 8.80·37-s − 3.12·39-s + 2.43·41-s − 6.56·43-s − 8.24·47-s − 2.43·51-s + 1.12·53-s + 5.56·57-s − 11.3·59-s − 11.1·61-s − 7.87·67-s + 2.24·69-s + 1.68·71-s + 6.43·73-s − 5.68·79-s − 7·81-s + ⋯
L(s)  = 1  − 0.901·3-s − 0.187·9-s − 1.84·11-s + 0.554·13-s + 0.378·17-s − 0.817·19-s − 0.299·23-s + 1.07·27-s + 0.638·29-s + 1.63·31-s + 1.66·33-s − 1.44·37-s − 0.500·39-s + 0.380·41-s − 1.00·43-s − 1.20·47-s − 0.341·51-s + 0.154·53-s + 0.736·57-s − 1.48·59-s − 1.42·61-s − 0.962·67-s + 0.270·69-s + 0.199·71-s + 0.753·73-s − 0.639·79-s − 0.777·81-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

\(L(1)\) \(\approx\) \(0.6083163865\)
\(L(\frac12)\) \(\approx\) \(0.6083163865\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 - 9.12T + 31T^{2} \)
37 \( 1 + 8.80T + 37T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 + 6.56T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 1.12T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 7.87T + 67T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 - 6.43T + 73T^{2} \)
79 \( 1 + 5.68T + 79T^{2} \)
83 \( 1 + 1.31T + 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.77754541801266973320485513574, −6.81028084083697954922084973542, −6.21462683182755426618202483386, −5.66456427048430994745195336638, −4.94703115437678994228897046944, −4.51353252517580402730714679205, −3.25190110829429909570949590966, −2.72155501542962927823563927677, −1.61921784933775463351154997502, −0.38207870899316422186901465476, 0.38207870899316422186901465476, 1.61921784933775463351154997502, 2.72155501542962927823563927677, 3.25190110829429909570949590966, 4.51353252517580402730714679205, 4.94703115437678994228897046944, 5.66456427048430994745195336638, 6.21462683182755426618202483386, 6.81028084083697954922084973542, 7.77754541801266973320485513574

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