Properties

Label 2-6160-1.1-c1-0-32
Degree $2$
Conductor $6160$
Sign $1$
Analytic cond. $49.1878$
Root an. cond. $7.01340$
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  − 2·3-s + 5-s − 7-s + 9-s + 11-s + 6.74·13-s − 2·15-s − 6.74·17-s + 6.74·19-s + 2·21-s + 6.74·23-s + 25-s + 4·27-s + 8.74·29-s − 4.74·31-s − 2·33-s − 35-s + 0.744·37-s − 13.4·39-s − 4·41-s − 4·43-s + 45-s − 4.74·47-s + 49-s + 13.4·51-s − 12.7·53-s + 55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.87·13-s − 0.516·15-s − 1.63·17-s + 1.54·19-s + 0.436·21-s + 1.40·23-s + 0.200·25-s + 0.769·27-s + 1.62·29-s − 0.852·31-s − 0.348·33-s − 0.169·35-s + 0.122·37-s − 2.15·39-s − 0.624·41-s − 0.609·43-s + 0.149·45-s − 0.692·47-s + 0.142·49-s + 1.88·51-s − 1.75·53-s + 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6160\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(49.1878\)
Root analytic conductor: \(7.01340\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.474532359\)
\(L(\frac12)\) \(\approx\) \(1.474532359\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
good3 \( 1 + 2T + 3T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 6.74T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 - 6.74T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 - 0.744T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 - 8.74T + 59T^{2} \)
61 \( 1 - 1.25T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−8.186274701716392366393813704598, −6.79635309634068407321094202521, −6.71272381858003560925133403697, −5.99315221064332422531579841511, −5.25649386679768376749062680633, −4.68941108682314028770906250511, −3.61066040592259378630563523080, −2.88691340112206923875506277086, −1.52903130194778775408956197135, −0.72656273358875899614645617462, 0.72656273358875899614645617462, 1.52903130194778775408956197135, 2.88691340112206923875506277086, 3.61066040592259378630563523080, 4.68941108682314028770906250511, 5.25649386679768376749062680633, 5.99315221064332422531579841511, 6.71272381858003560925133403697, 6.79635309634068407321094202521, 8.186274701716392366393813704598

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