Properties

Label 2-5445-1.1-c1-0-110
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.456·2-s − 1.79·4-s + 5-s − 4.37·7-s − 1.73·8-s + 0.456·10-s − 0.913·13-s − 1.99·14-s + 2.79·16-s + 1.73·17-s + 3.46·19-s − 1.79·20-s + 0.582·23-s + 25-s − 0.417·26-s + 7.84·28-s + 9.66·29-s − 8.58·31-s + 4.73·32-s + 0.791·34-s − 4.37·35-s + 7.58·37-s + 1.58·38-s − 1.73·40-s + 3.46·41-s + 9.66·43-s + 0.266·46-s + ⋯
L(s)  = 1  + 0.323·2-s − 0.895·4-s + 0.447·5-s − 1.65·7-s − 0.612·8-s + 0.144·10-s − 0.253·13-s − 0.534·14-s + 0.697·16-s + 0.420·17-s + 0.794·19-s − 0.400·20-s + 0.121·23-s + 0.200·25-s − 0.0818·26-s + 1.48·28-s + 1.79·29-s − 1.54·31-s + 0.837·32-s + 0.135·34-s − 0.739·35-s + 1.24·37-s + 0.256·38-s − 0.273·40-s + 0.541·41-s + 1.47·43-s + 0.0392·46-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: \(1\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - 0.456T + 2T^{2} \)
7 \( 1 + 4.37T + 7T^{2} \)
13 \( 1 + 0.913T + 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 0.582T + 23T^{2} \)
29 \( 1 - 9.66T + 29T^{2} \)
31 \( 1 + 8.58T + 31T^{2} \)
37 \( 1 - 7.58T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 9.66T + 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 5.10T + 73T^{2} \)
79 \( 1 + 0.818T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 4.41T + 97T^{2} \)
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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

−7.75448580751075856928395803705, −7.03652833996812934370584313720, −6.05917868455455589890376678266, −5.84991854549980199846658892390, −4.84853783878581449964537635819, −4.12211849485841805392479364446, −3.15742600485573532902268378871, −2.80537086846802370294437404748, −1.15471173239259905997156054136, 0, 1.15471173239259905997156054136, 2.80537086846802370294437404748, 3.15742600485573532902268378871, 4.12211849485841805392479364446, 4.84853783878581449964537635819, 5.84991854549980199846658892390, 6.05917868455455589890376678266, 7.03652833996812934370584313720, 7.75448580751075856928395803705

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