Properties

Label 2-5520-1.1-c1-0-31
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 + 0.944·7-s + 9-s − 5.62·11-s + 4.42·13-s + 15-s + 3.41·17-s + 2.35·19-s + 0.944·21-s + 23-s + 25-s + 27-s + 8.57·29-s + 8.99·31-s − 5.62·33-s + 0.944·35-s + 1.05·37-s + 4.42·39-s − 9.35·41-s − 4.78·43-s + 45-s − 2.35·47-s − 6.10·49-s + 3.41·51-s − 11.4·53-s − 5.62·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.357·7-s + 0.333·9-s − 1.69·11-s + 1.22·13-s + 0.258·15-s + 0.828·17-s + 0.541·19-s + 0.206·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.59·29-s + 1.61·31-s − 0.979·33-s + 0.159·35-s + 0.173·37-s + 0.708·39-s − 1.46·41-s − 0.729·43-s + 0.149·45-s − 0.344·47-s − 0.872·49-s + 0.478·51-s − 1.57·53-s − 0.758·55-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

\(L(1)\) \(\approx\) \(2.978808477\)
\(L(\frac12)\) \(\approx\) \(2.978808477\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.944T + 7T^{2} \)
11 \( 1 + 5.62T + 11T^{2} \)
13 \( 1 - 4.42T + 13T^{2} \)
17 \( 1 - 3.41T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
29 \( 1 - 8.57T + 29T^{2} \)
31 \( 1 - 8.99T + 31T^{2} \)
37 \( 1 - 1.05T + 37T^{2} \)
41 \( 1 + 9.35T + 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 8.93T + 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 - 1.53T + 73T^{2} \)
79 \( 1 + 16.7T + 79T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 - 18.8T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−8.189101527392213688937917858857, −7.72707615414514505671458753513, −6.67045380705863239439292027622, −6.06685960428450268933182335323, −5.07209862188576399072850917395, −4.73514139854870051191566274948, −3.34512940301561918211363443646, −2.97150932880094719261851313122, −1.91315411310715533294463351270, −0.937127878183543181896950917617, 0.937127878183543181896950917617, 1.91315411310715533294463351270, 2.97150932880094719261851313122, 3.34512940301561918211363443646, 4.73514139854870051191566274948, 5.07209862188576399072850917395, 6.06685960428450268933182335323, 6.67045380705863239439292027622, 7.72707615414514505671458753513, 8.189101527392213688937917858857

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