Properties

Label 2-5445-1.1-c1-0-5
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  − 0.618·2-s − 1.61·4-s + 5-s − 3.23·7-s + 2.23·8-s − 0.618·10-s − 5.23·13-s + 2.00·14-s + 1.85·16-s − 5.47·17-s − 6.47·19-s − 1.61·20-s + 4.70·23-s + 25-s + 3.23·26-s + 5.23·28-s − 1.23·29-s − 6.70·31-s − 5.61·32-s + 3.38·34-s − 3.23·35-s + 0.763·37-s + 4.00·38-s + 2.23·40-s + 3.52·41-s + 5.23·43-s − 2.90·46-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s + 0.447·5-s − 1.22·7-s + 0.790·8-s − 0.195·10-s − 1.45·13-s + 0.534·14-s + 0.463·16-s − 1.32·17-s − 1.48·19-s − 0.361·20-s + 0.981·23-s + 0.200·25-s + 0.634·26-s + 0.989·28-s − 0.229·29-s − 1.20·31-s − 0.993·32-s + 0.580·34-s − 0.546·35-s + 0.125·37-s + 0.648·38-s + 0.353·40-s + 0.550·41-s + 0.798·43-s − 0.429·46-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3719663722\)
\(L(\frac12)\) \(\approx\) \(0.3719663722\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
7 \( 1 + 3.23T + 7T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 + 1.23T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 0.763T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + 8.70T + 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 1.47T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−8.324387151799127760147034121650, −7.43896861292988426411055241229, −6.75654219947332354995562589568, −6.17076113731847744224041122083, −5.13559460185910387870083039428, −4.58430576136809789874024714360, −3.73522973404937058473185919824, −2.72061068079549631747266905345, −1.88819327925384060815489269683, −0.33425767859809056623056024827, 0.33425767859809056623056024827, 1.88819327925384060815489269683, 2.72061068079549631747266905345, 3.73522973404937058473185919824, 4.58430576136809789874024714360, 5.13559460185910387870083039428, 6.17076113731847744224041122083, 6.75654219947332354995562589568, 7.43896861292988426411055241229, 8.324387151799127760147034121650

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