Properties

Label 2-5445-1.1-c1-0-128
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  − 1.61·2-s + 0.618·4-s + 5-s + 2·7-s + 2.23·8-s − 1.61·10-s − 6.47·13-s − 3.23·14-s − 4.85·16-s + 5.47·17-s + 0.618·20-s − 8.23·23-s + 25-s + 10.4·26-s + 1.23·28-s + 0.472·29-s + 6.70·31-s + 3.38·32-s − 8.85·34-s + 2·35-s + 8.47·37-s + 2.23·40-s − 6·41-s + 6·43-s + 13.3·46-s − 3.76·47-s − 3·49-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s + 0.447·5-s + 0.755·7-s + 0.790·8-s − 0.511·10-s − 1.79·13-s − 0.864·14-s − 1.21·16-s + 1.32·17-s + 0.138·20-s − 1.71·23-s + 0.200·25-s + 2.05·26-s + 0.233·28-s + 0.0876·29-s + 1.20·31-s + 0.597·32-s − 1.51·34-s + 0.338·35-s + 1.39·37-s + 0.353·40-s − 0.937·41-s + 0.914·43-s + 1.96·46-s − 0.549·47-s − 0.428·49-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 \( 1 + 1.61T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 - 0.472T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 3.76T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 5.47T + 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + 4.47T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 + 6.47T + 97T^{2} \)
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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

−7.85104762007924134326924273544, −7.55417391347621500210706945623, −6.53859738004114847533779928755, −5.67788507640243167994929001880, −4.81300267822563306392020736180, −4.35857974439496194957888368654, −2.95499680255394073791608234608, −2.05076328308005885778178367629, −1.26155695978226689010661198390, 0, 1.26155695978226689010661198390, 2.05076328308005885778178367629, 2.95499680255394073791608234608, 4.35857974439496194957888368654, 4.81300267822563306392020736180, 5.67788507640243167994929001880, 6.53859738004114847533779928755, 7.55417391347621500210706945623, 7.85104762007924134326924273544

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