Properties

Label 2-6160-1.1-c1-0-104
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  + 0.732·3-s + 5-s + 7-s − 2.46·9-s − 11-s + 0.347·13-s + 0.732·15-s + 2.60·17-s − 6.03·19-s + 0.732·21-s + 1.08·23-s + 25-s − 4·27-s − 9.49·29-s + 4.76·31-s − 0.732·33-s + 35-s − 4.28·37-s + 0.254·39-s + 9.84·41-s − 2.69·43-s − 2.46·45-s − 4.86·47-s + 49-s + 1.90·51-s − 3.77·53-s − 55-s + ⋯
L(s)  = 1  + 0.422·3-s + 0.447·5-s + 0.377·7-s − 0.821·9-s − 0.301·11-s + 0.0965·13-s + 0.189·15-s + 0.631·17-s − 1.38·19-s + 0.159·21-s + 0.225·23-s + 0.200·25-s − 0.769·27-s − 1.76·29-s + 0.855·31-s − 0.127·33-s + 0.169·35-s − 0.704·37-s + 0.0407·39-s + 1.53·41-s − 0.411·43-s − 0.367·45-s − 0.709·47-s + 0.142·49-s + 0.266·51-s − 0.518·53-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 \( 1 - 0.732T + 3T^{2} \)
13 \( 1 - 0.347T + 13T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
19 \( 1 + 6.03T + 19T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 - 4.76T + 31T^{2} \)
37 \( 1 + 4.28T + 37T^{2} \)
41 \( 1 - 9.84T + 41T^{2} \)
43 \( 1 + 2.69T + 43T^{2} \)
47 \( 1 + 4.86T + 47T^{2} \)
53 \( 1 + 3.77T + 53T^{2} \)
59 \( 1 + 2.18T + 59T^{2} \)
61 \( 1 + 0.883T + 61T^{2} \)
67 \( 1 + 8.95T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 + 6.54T + 79T^{2} \)
83 \( 1 + 4.76T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 2.12T + 97T^{2} \)
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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

−7.86040621175693758763490658642, −7.07986216839089624067782205787, −6.12160151212382129149174245581, −5.67881411363630386137277757862, −4.85102736200758706863113619467, −3.99020517254815306662395327681, −3.09552518809499059259569221930, −2.35265740038730560656642361828, −1.50895182621960977591609960772, 0, 1.50895182621960977591609960772, 2.35265740038730560656642361828, 3.09552518809499059259569221930, 3.99020517254815306662395327681, 4.85102736200758706863113619467, 5.67881411363630386137277757862, 6.12160151212382129149174245581, 7.07986216839089624067782205787, 7.86040621175693758763490658642

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