Properties

Label 2-5445-1.1-c1-0-100
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  − 2.16·2-s + 2.68·4-s − 5-s + 0.480·7-s − 1.48·8-s + 2.16·10-s + 3.36·13-s − 1.03·14-s − 2.16·16-s − 4.84·17-s − 1.84·19-s − 2.68·20-s − 3.48·23-s + 25-s − 7.28·26-s + 1.28·28-s + 8.32·29-s − 2.36·31-s + 7.64·32-s + 10.4·34-s − 0.480·35-s + 5.44·37-s + 4.00·38-s + 1.48·40-s + 6.65·41-s + 4.32·43-s + 7.53·46-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.34·4-s − 0.447·5-s + 0.181·7-s − 0.523·8-s + 0.684·10-s + 0.934·13-s − 0.277·14-s − 0.541·16-s − 1.17·17-s − 0.424·19-s − 0.600·20-s − 0.725·23-s + 0.200·25-s − 1.42·26-s + 0.243·28-s + 1.54·29-s − 0.425·31-s + 1.35·32-s + 1.79·34-s − 0.0811·35-s + 0.894·37-s + 0.648·38-s + 0.234·40-s + 1.03·41-s + 0.660·43-s + 1.11·46-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 \( 1 + 2.16T + 2T^{2} \)
7 \( 1 - 0.480T + 7T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 + 4.84T + 17T^{2} \)
19 \( 1 + 1.84T + 19T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 + 2.36T + 31T^{2} \)
37 \( 1 - 5.44T + 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 - 3.17T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 5.84T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 5.67T + 89T^{2} \)
97 \( 1 + 8.17T + 97T^{2} \)
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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

−8.073562548677035199130545632806, −7.39055940828771813588374384011, −6.48540622135389614232643070276, −6.11947815596001370720587612679, −4.69071472236870826497522126901, −4.19007627871217970029911206177, −2.98409083982161175448932903935, −2.03673627735786810172951185573, −1.10591055612874239596776060531, 0, 1.10591055612874239596776060531, 2.03673627735786810172951185573, 2.98409083982161175448932903935, 4.19007627871217970029911206177, 4.69071472236870826497522126901, 6.11947815596001370720587612679, 6.48540622135389614232643070276, 7.39055940828771813588374384011, 8.073562548677035199130545632806

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