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Properties

Label 2-221760-1.1-c1-0-70
Degree $2$
Conductor $221760$
Sign $1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
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  + 5-s − 7-s − 11-s + 6·13-s + 7·17-s − 5·19-s − 23-s + 25-s − 5·29-s + 8·31-s − 35-s + 2·37-s − 12·41-s − 11·43-s + 8·47-s + 49-s − 11·53-s − 55-s + 5·59-s − 7·61-s + 6·65-s − 2·67-s + 12·71-s + 4·73-s + 77-s + 10·79-s + 83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.66·13-s + 1.69·17-s − 1.14·19-s − 0.208·23-s + 1/5·25-s − 0.928·29-s + 1.43·31-s − 0.169·35-s + 0.328·37-s − 1.87·41-s − 1.67·43-s + 1.16·47-s + 1/7·49-s − 1.51·53-s − 0.134·55-s + 0.650·59-s − 0.896·61-s + 0.744·65-s − 0.244·67-s + 1.42·71-s + 0.468·73-s + 0.113·77-s + 1.12·79-s + 0.109·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.708133967\)
\(L(\frac12)\) \(\approx\) \(2.708133967\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.14183444961001, −12.45384101175585, −12.14620834455480, −11.56110212676976, −11.04143167099295, −10.58022754287976, −10.17646810045237, −9.750081234571284, −9.297415188307352, −8.616866310274127, −8.166793079054436, −8.050273009583157, −7.145131911524461, −6.675663277890183, −6.113098758474061, −5.935402416783696, −5.241771259477310, −4.780188183232273, −4.029641891598291, −3.438405467509732, −3.239982129895584, −2.397924958678321, −1.718542990219506, −1.239305911293598, −0.4719663578797958, 0.4719663578797958, 1.239305911293598, 1.718542990219506, 2.397924958678321, 3.239982129895584, 3.438405467509732, 4.029641891598291, 4.780188183232273, 5.241771259477310, 5.935402416783696, 6.113098758474061, 6.675663277890183, 7.145131911524461, 8.050273009583157, 8.166793079054436, 8.616866310274127, 9.297415188307352, 9.750081234571284, 10.17646810045237, 10.58022754287976, 11.04143167099295, 11.56110212676976, 12.14620834455480, 12.45384101175585, 13.14183444961001

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