Properties

Label 2-5445-1.1-c1-0-124
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  + 0.262·2-s − 1.93·4-s + 5-s − 0.704·7-s − 1.03·8-s + 0.262·10-s − 3.82·13-s − 0.185·14-s + 3.59·16-s + 7.15·17-s − 7.33·19-s − 1.93·20-s + 5.86·23-s + 25-s − 1.00·26-s + 1.36·28-s − 8.97·29-s − 0.590·31-s + 3.00·32-s + 1.88·34-s − 0.704·35-s + 10.4·37-s − 1.92·38-s − 1.03·40-s + 7.92·41-s − 3.29·43-s + 1.53·46-s + ⋯
L(s)  = 1  + 0.185·2-s − 0.965·4-s + 0.447·5-s − 0.266·7-s − 0.365·8-s + 0.0830·10-s − 1.05·13-s − 0.0494·14-s + 0.897·16-s + 1.73·17-s − 1.68·19-s − 0.431·20-s + 1.22·23-s + 0.200·25-s − 0.196·26-s + 0.257·28-s − 1.66·29-s − 0.106·31-s + 0.531·32-s + 0.322·34-s − 0.119·35-s + 1.71·37-s − 0.312·38-s − 0.163·40-s + 1.23·41-s − 0.502·43-s + 0.227·46-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 \( 1 - 0.262T + 2T^{2} \)
7 \( 1 + 0.704T + 7T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - 7.15T + 17T^{2} \)
19 \( 1 + 7.33T + 19T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 + 0.590T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 7.92T + 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 - 3.64T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 5.64T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 3.82T + 73T^{2} \)
79 \( 1 - 3.33T + 79T^{2} \)
83 \( 1 - 4.04T + 83T^{2} \)
89 \( 1 + 7.05T + 89T^{2} \)
97 \( 1 - 6.81T + 97T^{2} \)
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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

−7.73865473779360923127971591301, −7.26131481999407680597032731509, −6.07311189815756246495054577259, −5.70186028638745848183985992337, −4.83518598487653456233887932841, −4.22944069519995366685130447278, −3.29716056899899781201555065536, −2.50359854118842285618899342756, −1.24399216576893884288465950771, 0, 1.24399216576893884288465950771, 2.50359854118842285618899342756, 3.29716056899899781201555065536, 4.22944069519995366685130447278, 4.83518598487653456233887932841, 5.70186028638745848183985992337, 6.07311189815756246495054577259, 7.26131481999407680597032731509, 7.73865473779360923127971591301

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