Properties

Label 2-6080-1.1-c1-0-32
Degree $2$
Conductor $6080$
Sign $1$
Analytic cond. $48.5490$
Root an. cond. $6.96771$
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  + 1.76·3-s + 5-s − 3.89·7-s + 0.114·9-s − 4.80·11-s − 0.0353·13-s + 1.76·15-s − 4.39·17-s − 19-s − 6.86·21-s + 9.12·23-s + 25-s − 5.09·27-s + 6.21·29-s + 9.71·31-s − 8.48·33-s − 3.89·35-s + 0.749·37-s − 0.0623·39-s − 3.52·41-s + 11.4·43-s + 0.114·45-s + 5.57·47-s + 8.15·49-s − 7.75·51-s − 8.67·53-s − 4.80·55-s + ⋯
L(s)  = 1  + 1.01·3-s + 0.447·5-s − 1.47·7-s + 0.0382·9-s − 1.44·11-s − 0.00979·13-s + 0.455·15-s − 1.06·17-s − 0.229·19-s − 1.49·21-s + 1.90·23-s + 0.200·25-s − 0.979·27-s + 1.15·29-s + 1.74·31-s − 1.47·33-s − 0.657·35-s + 0.123·37-s − 0.00997·39-s − 0.551·41-s + 1.74·43-s + 0.0171·45-s + 0.813·47-s + 1.16·49-s − 1.08·51-s − 1.19·53-s − 0.648·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6080\)    =    \(2^{6} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(48.5490\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.001214200\)
\(L(\frac12)\) \(\approx\) \(2.001214200\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + T \)
good3 \( 1 - 1.76T + 3T^{2} \)
7 \( 1 + 3.89T + 7T^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
13 \( 1 + 0.0353T + 13T^{2} \)
17 \( 1 + 4.39T + 17T^{2} \)
23 \( 1 - 9.12T + 23T^{2} \)
29 \( 1 - 6.21T + 29T^{2} \)
31 \( 1 - 9.71T + 31T^{2} \)
37 \( 1 - 0.749T + 37T^{2} \)
41 \( 1 + 3.52T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 + 8.67T + 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 + 6.50T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 5.64T + 83T^{2} \)
89 \( 1 + 1.24T + 89T^{2} \)
97 \( 1 - 9.31T + 97T^{2} \)
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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

−8.189798851762106049105541268096, −7.40212854268113117475961364545, −6.63757147678213269129641672663, −6.11191442501317406081049087362, −5.16128629256088675262469820050, −4.40548933699646528307352793461, −3.24203962183610341734728730455, −2.78710467386997000120806640909, −2.32279983149029754413364944079, −0.66516431469268541983840734608, 0.66516431469268541983840734608, 2.32279983149029754413364944079, 2.78710467386997000120806640909, 3.24203962183610341734728730455, 4.40548933699646528307352793461, 5.16128629256088675262469820050, 6.11191442501317406081049087362, 6.63757147678213269129641672663, 7.40212854268113117475961364545, 8.189798851762106049105541268096

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