Properties

Label 2-6160-1.1-c1-0-106
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 5-s − 7-s − 0.999·9-s + 11-s − 3.41·13-s + 1.41·15-s + 2.24·17-s − 5.65·19-s − 1.41·21-s + 2·23-s + 25-s − 5.65·27-s − 3.17·29-s + 4.58·31-s + 1.41·33-s − 35-s − 2.82·37-s − 4.82·39-s + 4.24·41-s + 2·43-s − 0.999·45-s + 8.24·47-s + 49-s + 3.17·51-s − 4.48·53-s + 55-s + ⋯
L(s)  = 1  + 0.816·3-s + 0.447·5-s − 0.377·7-s − 0.333·9-s + 0.301·11-s − 0.946·13-s + 0.365·15-s + 0.543·17-s − 1.29·19-s − 0.308·21-s + 0.417·23-s + 0.200·25-s − 1.08·27-s − 0.588·29-s + 0.823·31-s + 0.246·33-s − 0.169·35-s − 0.464·37-s − 0.773·39-s + 0.662·41-s + 0.304·43-s − 0.149·45-s + 1.20·47-s + 0.142·49-s + 0.444·51-s − 0.616·53-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6160,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 + 4.48T + 53T^{2} \)
59 \( 1 + 4.58T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 9.31T + 67T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 7.17T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + 3.17T + 89T^{2} \)
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

−7.68688458720070061845835381524, −7.15358351546254101976085321244, −6.20664932831447427557819146355, −5.71873819111444741285944103849, −4.71453486660785567636391838404, −3.97639587985772572970906002800, −2.97704764759380362073760594718, −2.52275961003715365304911276551, −1.52916152982208687347970503711, 0, 1.52916152982208687347970503711, 2.52275961003715365304911276551, 2.97704764759380362073760594718, 3.97639587985772572970906002800, 4.71453486660785567636391838404, 5.71873819111444741285944103849, 6.20664932831447427557819146355, 7.15358351546254101976085321244, 7.68688458720070061845835381524

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