Properties

Label 2-5445-1.1-c1-0-120
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  + 2.74·2-s + 5.53·4-s + 5-s − 2.32·7-s + 9.70·8-s + 2.74·10-s − 0.534·13-s − 6.37·14-s + 15.5·16-s − 2.42·17-s + 4.95·19-s + 5.53·20-s + 4.53·23-s + 25-s − 1.46·26-s − 12.8·28-s + 5.48·29-s + 1.04·31-s + 23.3·32-s − 6.64·34-s − 2.32·35-s + 7.48·37-s + 13.6·38-s + 9.70·40-s − 10.6·41-s + 4.32·43-s + 12.4·46-s + ⋯
L(s)  = 1  + 1.94·2-s + 2.76·4-s + 0.447·5-s − 0.878·7-s + 3.42·8-s + 0.867·10-s − 0.148·13-s − 1.70·14-s + 3.88·16-s − 0.587·17-s + 1.13·19-s + 1.23·20-s + 0.945·23-s + 0.200·25-s − 0.287·26-s − 2.42·28-s + 1.01·29-s + 0.187·31-s + 4.11·32-s − 1.13·34-s − 0.392·35-s + 1.23·37-s + 2.20·38-s + 1.53·40-s − 1.65·41-s + 0.659·43-s + 1.83·46-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

\(L(1)\) \(\approx\) \(7.942791935\)
\(L(\frac12)\) \(\approx\) \(7.942791935\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 2T^{2} \)
7 \( 1 + 2.32T + 7T^{2} \)
13 \( 1 + 0.534T + 13T^{2} \)
17 \( 1 + 2.42T + 17T^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 - 4.53T + 23T^{2} \)
29 \( 1 - 5.48T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 - 7.48T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 6.76T + 47T^{2} \)
53 \( 1 - 4.53T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 6.79T + 61T^{2} \)
67 \( 1 - 0.721T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 - 1.06T + 73T^{2} \)
79 \( 1 - 4.64T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 4.75T + 97T^{2} \)
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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

−7.80902454922909711563395048227, −6.92291377750628066980222872043, −6.63367595132886303036809652483, −5.87625177099600145551527276610, −5.18647817886039096147531553305, −4.61060549686317687504596087471, −3.68219980769820415269146728117, −2.99234518082974107646221834761, −2.42820340174775604730745395283, −1.22404629362519886224800263440, 1.22404629362519886224800263440, 2.42820340174775604730745395283, 2.99234518082974107646221834761, 3.68219980769820415269146728117, 4.61060549686317687504596087471, 5.18647817886039096147531553305, 5.87625177099600145551527276610, 6.63367595132886303036809652483, 6.92291377750628066980222872043, 7.80902454922909711563395048227

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