Properties

Label 2-5445-1.1-c1-0-139
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  + 2.69·2-s + 5.27·4-s − 5-s + 4.13·7-s + 8.82·8-s − 2.69·10-s − 2.73·13-s + 11.1·14-s + 13.2·16-s + 2.41·17-s + 1.15·19-s − 5.27·20-s + 8.54·23-s + 25-s − 7.38·26-s + 21.7·28-s − 1.67·29-s − 10.2·31-s + 18.1·32-s + 6.51·34-s − 4.13·35-s + 5.71·37-s + 3.11·38-s − 8.82·40-s − 9.11·41-s − 8.13·43-s + 23.0·46-s + ⋯
L(s)  = 1  + 1.90·2-s + 2.63·4-s − 0.447·5-s + 1.56·7-s + 3.12·8-s − 0.852·10-s − 0.759·13-s + 2.97·14-s + 3.31·16-s + 0.585·17-s + 0.264·19-s − 1.17·20-s + 1.78·23-s + 0.200·25-s − 1.44·26-s + 4.11·28-s − 0.311·29-s − 1.84·31-s + 3.20·32-s + 1.11·34-s − 0.698·35-s + 0.939·37-s + 0.504·38-s − 1.39·40-s − 1.42·41-s − 1.24·43-s + 3.39·46-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: $\chi_{5445} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.262726827\)
\(L(\frac12)\) \(\approx\) \(8.262726827\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.69T + 2T^{2} \)
7 \( 1 - 4.13T + 7T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 - 2.41T + 17T^{2} \)
19 \( 1 - 1.15T + 19T^{2} \)
23 \( 1 - 8.54T + 23T^{2} \)
29 \( 1 + 1.67T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 - 5.71T + 37T^{2} \)
41 \( 1 + 9.11T + 41T^{2} \)
43 \( 1 + 8.13T + 43T^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 + 2.73T + 73T^{2} \)
79 \( 1 + 5.15T + 79T^{2} \)
83 \( 1 - 9.28T + 83T^{2} \)
89 \( 1 + 4.78T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−7.81563112237739054306354184025, −7.23382685215008308241580991758, −6.75383116322746487919078331786, −5.47922591673372099419699497148, −5.21158547508398914567277208359, −4.65099194425477604175471530950, −3.81594575928137466314417948030, −3.10864741217229663290189037630, −2.15925475189847940063389887382, −1.32586236387447493654157080695, 1.32586236387447493654157080695, 2.15925475189847940063389887382, 3.10864741217229663290189037630, 3.81594575928137466314417948030, 4.65099194425477604175471530950, 5.21158547508398914567277208359, 5.47922591673372099419699497148, 6.75383116322746487919078331786, 7.23382685215008308241580991758, 7.81563112237739054306354184025

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