Properties

Label 2-920-1.1-c1-0-2
Degree $2$
Conductor $920$
Sign $1$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 2.14·3-s − 5-s − 1.14·7-s + 1.60·9-s + 5.89·11-s − 4.89·13-s + 2.14·15-s − 5.89·17-s − 2.34·19-s + 2.45·21-s + 23-s + 25-s + 3.00·27-s + 3.74·29-s + 5.68·31-s − 12.6·33-s + 1.14·35-s + 4·37-s + 10.4·39-s − 1.05·41-s + 11.4·43-s − 1.60·45-s + 7.74·47-s − 5.68·49-s + 12.6·51-s + 12.9·53-s − 5.89·55-s + ⋯
L(s)  = 1  − 1.23·3-s − 0.447·5-s − 0.432·7-s + 0.533·9-s + 1.77·11-s − 1.35·13-s + 0.553·15-s − 1.42·17-s − 0.538·19-s + 0.536·21-s + 0.208·23-s + 0.200·25-s + 0.577·27-s + 0.695·29-s + 1.02·31-s − 2.20·33-s + 0.193·35-s + 0.657·37-s + 1.68·39-s − 0.165·41-s + 1.75·43-s − 0.238·45-s + 1.12·47-s − 0.812·49-s + 1.76·51-s + 1.78·53-s − 0.794·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7461607600\)
\(L(\frac12)\) \(\approx\) \(0.7461607600\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + 2.14T + 3T^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 - 5.89T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
29 \( 1 - 3.74T + 29T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 0.797T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 2.94T + 71T^{2} \)
73 \( 1 - 6.32T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 0.912T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−10.20367433024115977059770968872, −9.296071747809153265298953774876, −8.549828905158112225631934999399, −7.14021441660917563835613240358, −6.65038450316429012628623732678, −5.90017753193875975243790397177, −4.66439229664051680915013797051, −4.11741112209726206191038807140, −2.52431916908950060007216188222, −0.71616426428189796732043876617, 0.71616426428189796732043876617, 2.52431916908950060007216188222, 4.11741112209726206191038807140, 4.66439229664051680915013797051, 5.90017753193875975243790397177, 6.65038450316429012628623732678, 7.14021441660917563835613240358, 8.549828905158112225631934999399, 9.296071747809153265298953774876, 10.20367433024115977059770968872

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