Properties

Label 2-5445-1.1-c1-0-28
Degree $2$
Conductor $5445$
Sign $1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
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  + 1.73·2-s + 0.999·4-s − 5-s − 3.46·7-s − 1.73·8-s − 1.73·10-s − 5.99·14-s − 5·16-s − 6.92·17-s + 6.92·19-s − 0.999·20-s − 6·23-s + 25-s − 3.46·28-s + 4·31-s − 5.19·32-s − 11.9·34-s + 3.46·35-s + 10·37-s + 11.9·38-s + 1.73·40-s + 6.92·41-s + 3.46·43-s − 10.3·46-s + 6·47-s + 4.99·49-s + 1.73·50-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s − 0.447·5-s − 1.30·7-s − 0.612·8-s − 0.547·10-s − 1.60·14-s − 1.25·16-s − 1.68·17-s + 1.58·19-s − 0.223·20-s − 1.25·23-s + 0.200·25-s − 0.654·28-s + 0.718·31-s − 0.918·32-s − 2.05·34-s + 0.585·35-s + 1.64·37-s + 1.94·38-s + 0.273·40-s + 1.08·41-s + 0.528·43-s − 1.53·46-s + 0.875·47-s + 0.714·49-s + 0.244·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.4785\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.947789028\)
\(L(\frac12)\) \(\approx\) \(1.947789028\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.027636892918290837055807022337, −7.23069055659056595354230287574, −6.45262601060128533726689027701, −6.03672883682931054608127556513, −5.22536395464530243749868636441, −4.24887214931824173680318543131, −3.93968158247696526365783279981, −2.96643778208936220411499732121, −2.43190627109958572401855629163, −0.59033297671018356870742448008, 0.59033297671018356870742448008, 2.43190627109958572401855629163, 2.96643778208936220411499732121, 3.93968158247696526365783279981, 4.24887214931824173680318543131, 5.22536395464530243749868636441, 6.03672883682931054608127556513, 6.45262601060128533726689027701, 7.23069055659056595354230287574, 8.027636892918290837055807022337

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