Properties

Label 2-5544-1.1-c1-0-21
Degree $2$
Conductor $5544$
Sign $1$
Analytic cond. $44.2690$
Root an. cond. $6.65350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 11-s + 4·13-s + 6·17-s + 2·19-s + 2·23-s − 25-s + 2·29-s + 8·31-s − 2·35-s − 2·37-s − 6·41-s − 2·43-s − 10·47-s + 49-s − 14·53-s − 2·55-s + 12·61-s − 8·65-s − 8·67-s + 6·71-s + 6·73-s + 77-s + 4·79-s + 4·83-s − 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s − 1.45·47-s + 1/7·49-s − 1.92·53-s − 0.269·55-s + 1.53·61-s − 0.992·65-s − 0.977·67-s + 0.712·71-s + 0.702·73-s + 0.113·77-s + 0.450·79-s + 0.439·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5544\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(44.2690\)
Root analytic conductor: \(6.65350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5544} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5544,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.989204282\)
\(L(\frac12)\) \(\approx\) \(1.989204282\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.253870118047246049301655816233, −7.57449588461501117278574540894, −6.74984743822073141357576007866, −6.05645528416025818903811470667, −5.17104911383420608330422415529, −4.48689191843270072600694940256, −3.52209403073156167642921466828, −3.16748484239402791528516981191, −1.67215698911536160234018417205, −0.806231245703167526414058551624, 0.806231245703167526414058551624, 1.67215698911536160234018417205, 3.16748484239402791528516981191, 3.52209403073156167642921466828, 4.48689191843270072600694940256, 5.17104911383420608330422415529, 6.05645528416025818903811470667, 6.74984743822073141357576007866, 7.57449588461501117278574540894, 8.253870118047246049301655816233

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