Properties

Label 2-70e2-1.1-c1-0-18
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s + 6·11-s − 2·13-s + 6·17-s + 8·19-s − 3·23-s + 5·27-s + 3·29-s + 2·31-s − 6·33-s − 8·37-s + 2·39-s − 3·41-s − 5·43-s − 6·51-s − 12·53-s − 8·57-s − 61-s + 7·67-s + 3·69-s + 10·73-s − 4·79-s + 81-s − 3·83-s − 3·87-s − 3·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 1.80·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 0.962·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s − 0.468·41-s − 0.762·43-s − 0.840·51-s − 1.64·53-s − 1.05·57-s − 0.128·61-s + 0.855·67-s + 0.361·69-s + 1.17·73-s − 0.450·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s − 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713924759\)
\(L(\frac12)\) \(\approx\) \(1.713924759\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−8.255068422866006050104579822444, −7.47412515521946972887964479492, −6.72955758821946826369657074289, −6.08726621013549838366168527268, −5.35906560962029520584328319073, −4.75436843540763875036298944584, −3.56939663785267587175216764450, −3.12993958273370302783212199139, −1.67317137806908499175125329209, −0.78409287570769326854101260467, 0.78409287570769326854101260467, 1.67317137806908499175125329209, 3.12993958273370302783212199139, 3.56939663785267587175216764450, 4.75436843540763875036298944584, 5.35906560962029520584328319073, 6.08726621013549838366168527268, 6.72955758821946826369657074289, 7.47412515521946972887964479492, 8.255068422866006050104579822444

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