Properties

Label 2-2900-1.1-c1-0-13
Degree $2$
Conductor $2900$
Sign $1$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
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  + 2·7-s − 3·9-s − 4·11-s + 6·13-s + 4·17-s + 4·19-s − 6·23-s − 29-s + 8·37-s − 2·41-s − 4·43-s + 4·47-s − 3·49-s + 2·53-s + 8·59-s + 10·61-s − 6·63-s + 10·67-s − 8·71-s − 8·77-s + 8·79-s + 9·81-s + 6·83-s + 6·89-s + 12·91-s + 12·97-s + 12·99-s + ⋯
L(s)  = 1  + 0.755·7-s − 9-s − 1.20·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s − 0.185·29-s + 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s + 1.04·59-s + 1.28·61-s − 0.755·63-s + 1.22·67-s − 0.949·71-s − 0.911·77-s + 0.900·79-s + 81-s + 0.658·83-s + 0.635·89-s + 1.25·91-s + 1.21·97-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.877332671\)
\(L(\frac12)\) \(\approx\) \(1.877332671\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−8.437982612948757466144236967999, −8.166344273072527605155220037306, −7.52041515223519037401787250034, −6.28804970205735497830422095583, −5.62139491694076595892476908285, −5.12393318973471119896749759750, −3.90459166688862309008139341358, −3.13872298844776834415772073961, −2.10194821002577286786817841759, −0.856899419421507899299452669904, 0.856899419421507899299452669904, 2.10194821002577286786817841759, 3.13872298844776834415772073961, 3.90459166688862309008139341358, 5.12393318973471119896749759750, 5.62139491694076595892476908285, 6.28804970205735497830422095583, 7.52041515223519037401787250034, 8.166344273072527605155220037306, 8.437982612948757466144236967999

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