Properties

Label 2-44880-1.1-c1-0-14
Degree $2$
Conductor $44880$
Sign $1$
Analytic cond. $358.368$
Root an. cond. $18.9306$
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  + 3-s − 5-s + 2·7-s + 9-s − 11-s − 15-s + 17-s + 2·21-s + 4·23-s + 25-s + 27-s + 2·29-s − 4·31-s − 33-s − 2·35-s − 2·37-s + 6·43-s − 45-s − 3·49-s + 51-s − 6·53-s + 55-s − 14·59-s − 2·61-s + 2·63-s − 14·67-s + 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.242·17-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.338·35-s − 0.328·37-s + 0.914·43-s − 0.149·45-s − 3/7·49-s + 0.140·51-s − 0.824·53-s + 0.134·55-s − 1.82·59-s − 0.256·61-s + 0.251·63-s − 1.71·67-s + 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(358.368\)
Root analytic conductor: \(18.9306\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{44880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 44880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.835519777\)
\(L(\frac12)\) \(\approx\) \(2.835519777\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.61285679606365, −14.10383373003654, −13.84587611257092, −13.03628726309776, −12.58785802011338, −12.20964793462270, −11.37809455769008, −11.08441900522350, −10.56611920291475, −9.901189269405759, −9.323631288929004, −8.735523534643315, −8.366918253517864, −7.563570993919229, −7.514532350779639, −6.719333290491176, −5.969622265714366, −5.344860743255107, −4.565766573960281, −4.394305308547892, −3.249493518677046, −3.124908862613711, −2.081618955394070, −1.511508944310342, −0.5880519075594078, 0.5880519075594078, 1.511508944310342, 2.081618955394070, 3.124908862613711, 3.249493518677046, 4.394305308547892, 4.565766573960281, 5.344860743255107, 5.969622265714366, 6.719333290491176, 7.514532350779639, 7.563570993919229, 8.366918253517864, 8.735523534643315, 9.323631288929004, 9.901189269405759, 10.56611920291475, 11.08441900522350, 11.37809455769008, 12.20964793462270, 12.58785802011338, 13.03628726309776, 13.84587611257092, 14.10383373003654, 14.61285679606365

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