Properties

Label 2-70e2-1.1-c1-0-22
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s − 5.27·11-s + 2.62·13-s + 0.418·17-s − 3.27·19-s + 7.82·23-s − 5.19·27-s + 4.27·29-s − 3.27·31-s − 9.13·33-s + 9.97·37-s + 4.54·39-s + 3.72·41-s + 2.15·43-s + 6.50·47-s + 0.725·51-s + 5.67·53-s − 5.67·57-s + 3.27·59-s + 13.5·61-s − 3.52·67-s + 13.5·69-s − 4.54·71-s + 6.50·73-s + 7.27·79-s − 9·81-s − 7.40·83-s + ⋯
L(s)  = 1  + 1.00·3-s − 1.59·11-s + 0.728·13-s + 0.101·17-s − 0.751·19-s + 1.63·23-s − 1.00·27-s + 0.793·29-s − 0.588·31-s − 1.59·33-s + 1.63·37-s + 0.728·39-s + 0.581·41-s + 0.327·43-s + 0.949·47-s + 0.101·51-s + 0.779·53-s − 0.751·57-s + 0.426·59-s + 1.73·61-s − 0.430·67-s + 1.63·69-s − 0.539·71-s + 0.761·73-s + 0.818·79-s − 81-s − 0.812·83-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: no
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500562249\)
\(L(\frac12)\) \(\approx\) \(2.500562249\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 0.418T + 17T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 - 7.82T + 23T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 - 9.97T + 37T^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 - 2.15T + 43T^{2} \)
47 \( 1 - 6.50T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 - 3.27T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 3.52T + 67T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 - 7.27T + 79T^{2} \)
83 \( 1 + 7.40T + 83T^{2} \)
89 \( 1 - 7T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.326121027905072772811713769516, −7.70095685752843726620659018585, −7.03260177340731552516444411941, −6.01386146689088099326891195109, −5.36470254792710437397608647081, −4.47715449527884593327618765704, −3.57392938751520457393356303725, −2.73959997294310563654271200821, −2.28591405637713447536171358815, −0.812578734995134305732650201349, 0.812578734995134305732650201349, 2.28591405637713447536171358815, 2.73959997294310563654271200821, 3.57392938751520457393356303725, 4.47715449527884593327618765704, 5.36470254792710437397608647081, 6.01386146689088099326891195109, 7.03260177340731552516444411941, 7.70095685752843726620659018585, 8.326121027905072772811713769516

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