Properties

Label 2-90e2-1.1-c1-0-25
Degree $2$
Conductor $8100$
Sign $1$
Analytic cond. $64.6788$
Root an. cond. $8.04231$
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  + 1.28·7-s + 4.14·11-s − 6.52·13-s + 5.98·17-s + 7.17·19-s − 7.53·23-s + 5.19·29-s + 5.17·31-s − 5.24·37-s + 0.680·41-s + 1.28·43-s + 5.31·47-s − 5.35·49-s + 2.21·53-s − 7.60·59-s − 2.17·61-s + 15.6·67-s + 5.50·71-s + 7.80·73-s + 5.31·77-s + 6·79-s − 9.75·83-s − 10.0·89-s − 8.35·91-s − 14.3·97-s − 0.680·101-s + 2.56·103-s + ⋯
L(s)  = 1  + 0.484·7-s + 1.24·11-s − 1.80·13-s + 1.45·17-s + 1.64·19-s − 1.57·23-s + 0.964·29-s + 0.930·31-s − 0.861·37-s + 0.106·41-s + 0.195·43-s + 0.774·47-s − 0.765·49-s + 0.304·53-s − 0.990·59-s − 0.278·61-s + 1.90·67-s + 0.653·71-s + 0.913·73-s + 0.605·77-s + 0.675·79-s − 1.07·83-s − 1.06·89-s − 0.876·91-s − 1.45·97-s − 0.0677·101-s + 0.252·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(64.6788\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.392237367\)
\(L(\frac12)\) \(\approx\) \(2.392237367\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 - 4.14T + 11T^{2} \)
13 \( 1 + 6.52T + 13T^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 - 7.17T + 19T^{2} \)
23 \( 1 + 7.53T + 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 - 5.17T + 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 - 0.680T + 41T^{2} \)
43 \( 1 - 1.28T + 43T^{2} \)
47 \( 1 - 5.31T + 47T^{2} \)
53 \( 1 - 2.21T + 53T^{2} \)
59 \( 1 + 7.60T + 59T^{2} \)
61 \( 1 + 2.17T + 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 - 5.50T + 71T^{2} \)
73 \( 1 - 7.80T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 9.75T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−7.81726809178239650141351666000, −7.21165245388660287279042275023, −6.52547911154600433596491654035, −5.62317082763675908492760565503, −5.07498461678638488770860044933, −4.32648473179977133020832413293, −3.50135310893948864478474732861, −2.69431074963731290549138721701, −1.69485324592483555795211047971, −0.800509856616437075921373256101, 0.800509856616437075921373256101, 1.69485324592483555795211047971, 2.69431074963731290549138721701, 3.50135310893948864478474732861, 4.32648473179977133020832413293, 5.07498461678638488770860044933, 5.62317082763675908492760565503, 6.52547911154600433596491654035, 7.21165245388660287279042275023, 7.81726809178239650141351666000

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