Properties

Label 2-5520-1.1-c1-0-22
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 3.12·7-s + 9-s + 3.12·11-s + 2·13-s + 15-s − 1.12·17-s − 4·19-s − 3.12·21-s − 23-s + 25-s + 27-s + 2·29-s + 3.12·33-s − 3.12·35-s + 1.12·37-s + 2·39-s + 2·41-s + 45-s + 8·47-s + 2.75·49-s − 1.12·51-s + 12.2·53-s + 3.12·55-s − 4·57-s − 2.24·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.18·7-s + 0.333·9-s + 0.941·11-s + 0.554·13-s + 0.258·15-s − 0.272·17-s − 0.917·19-s − 0.681·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 0.543·33-s − 0.527·35-s + 0.184·37-s + 0.320·39-s + 0.312·41-s + 0.149·45-s + 1.16·47-s + 0.393·49-s − 0.157·51-s + 1.68·53-s + 0.421·55-s − 0.529·57-s − 0.292·59-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.469285115\)
\(L(\frac12)\) \(\approx\) \(2.469285115\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 2.24T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 + 3.12T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.378773464612410435369465851807, −7.33877564468612088455253059729, −6.57718184349462358510254592393, −6.27736588391756662698826127512, −5.36681555767448325692362727495, −4.16502675631748651459014362026, −3.75780652074364951223220137591, −2.79026582685072621342139630451, −2.02040657616326832427191301053, −0.820781935671373985353206185677, 0.820781935671373985353206185677, 2.02040657616326832427191301053, 2.79026582685072621342139630451, 3.75780652074364951223220137591, 4.16502675631748651459014362026, 5.36681555767448325692362727495, 6.27736588391756662698826127512, 6.57718184349462358510254592393, 7.33877564468612088455253059729, 8.378773464612410435369465851807

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