Properties

Label 2-5520-1.1-c1-0-28
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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 + 9-s + 6·13-s − 15-s + 2·17-s + 23-s + 25-s + 27-s + 6·29-s − 8·31-s + 10·37-s + 6·39-s − 6·41-s + 8·43-s − 45-s − 8·47-s − 7·49-s + 2·51-s − 6·53-s + 4·59-s − 6·61-s − 6·65-s − 8·67-s + 69-s + 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.744·65-s − 0.977·67-s + 0.120·69-s + 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.572657568\)
\(L(\frac12)\) \(\approx\) \(2.572657568\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.019359359297544311096761708925, −7.75393713467786547327847976436, −6.66317179399237957120169035073, −6.17639171616166515298521856012, −5.19973102520078041679919841676, −4.33942834181879546583004669500, −3.56996289387197501612160070732, −3.04148862989307749653136785216, −1.83359373015065893554420547885, −0.874778452891290069414683902721, 0.874778452891290069414683902721, 1.83359373015065893554420547885, 3.04148862989307749653136785216, 3.56996289387197501612160070732, 4.33942834181879546583004669500, 5.19973102520078041679919841676, 6.17639171616166515298521856012, 6.66317179399237957120169035073, 7.75393713467786547327847976436, 8.019359359297544311096761708925

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