Properties

Label 2-90e2-1.1-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79·7-s − 3.10·11-s − 2.79·13-s + 4.83·17-s + 3.58·19-s + 4.83·23-s − 3.46·29-s + 6.58·31-s + 3.37·37-s − 6.56·41-s − 2.79·43-s − 8.66·47-s + 0.791·49-s − 4.83·53-s + 8.29·59-s + 10.3·61-s − 5·67-s + 7.93·71-s − 5·73-s + 8.66·77-s − 4·79-s + 1.00·83-s + 13.1·89-s + 7.79·91-s + 1.16·97-s + 15.2·101-s + 3.37·103-s + ⋯
L(s)  = 1  − 1.05·7-s − 0.935·11-s − 0.774·13-s + 1.17·17-s + 0.821·19-s + 1.00·23-s − 0.643·29-s + 1.18·31-s + 0.554·37-s − 1.02·41-s − 0.425·43-s − 1.26·47-s + 0.113·49-s − 0.664·53-s + 1.08·59-s + 1.32·61-s − 0.610·67-s + 0.941·71-s − 0.585·73-s + 0.986·77-s − 0.450·79-s + 0.110·83-s + 1.39·89-s + 0.816·91-s + 0.118·97-s + 1.51·101-s + 0.332·103-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: \(1\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 + 2.79T + 13T^{2} \)
17 \( 1 - 4.83T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 - 4.83T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 - 3.37T + 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 + 2.79T + 43T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 + 4.83T + 53T^{2} \)
59 \( 1 - 8.29T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 - 7.93T + 71T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 1.00T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 1.16T + 97T^{2} \)
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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

−7.52863985017980875359213283878, −6.79689867816598873173365883789, −6.16199632449619707067936761853, −5.19807333911657091449074768315, −4.97266539799031985395548138860, −3.66227039264918717324518296356, −3.10366048796960411398341613972, −2.44890255371490578149139110767, −1.13256899798669181458079939632, 0, 1.13256899798669181458079939632, 2.44890255371490578149139110767, 3.10366048796960411398341613972, 3.66227039264918717324518296356, 4.97266539799031985395548138860, 5.19807333911657091449074768315, 6.16199632449619707067936761853, 6.79689867816598873173365883789, 7.52863985017980875359213283878

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