Properties

Label 2-6160-1.1-c1-0-75
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  − 1.21·3-s − 5-s + 7-s − 1.53·9-s + 11-s − 1.21·13-s + 1.21·15-s + 4.27·17-s − 7.06·19-s − 1.21·21-s + 2.95·23-s + 25-s + 5.48·27-s − 2·29-s + 6.16·31-s − 1.21·33-s − 35-s − 4.95·37-s + 1.46·39-s + 0.166·41-s − 6.44·43-s + 1.53·45-s + 2.70·47-s + 49-s − 5.18·51-s + 4.11·53-s − 55-s + ⋯
L(s)  = 1  − 0.699·3-s − 0.447·5-s + 0.377·7-s − 0.511·9-s + 0.301·11-s − 0.335·13-s + 0.312·15-s + 1.03·17-s − 1.62·19-s − 0.264·21-s + 0.616·23-s + 0.200·25-s + 1.05·27-s − 0.371·29-s + 1.10·31-s − 0.210·33-s − 0.169·35-s − 0.814·37-s + 0.234·39-s + 0.0259·41-s − 0.982·43-s + 0.228·45-s + 0.393·47-s + 0.142·49-s − 0.725·51-s + 0.564·53-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: \(1\)
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 \( 1 + 1.21T + 3T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 + 7.06T + 19T^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 + 4.95T + 37T^{2} \)
41 \( 1 - 0.166T + 41T^{2} \)
43 \( 1 + 6.44T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 4.11T + 53T^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 + 6.13T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 - 2.42T + 89T^{2} \)
97 \( 1 + 7.71T + 97T^{2} \)
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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

−7.72184729672029655294569433513, −6.88954559492186336378664122714, −6.31268869311300036962581402023, −5.50859115809739797985054492547, −4.90651111433906934923061444325, −4.14715839200490986527462785506, −3.25967460176208348156636914442, −2.32013302890219695624796532015, −1.12524496262892800779036826500, 0, 1.12524496262892800779036826500, 2.32013302890219695624796532015, 3.25967460176208348156636914442, 4.14715839200490986527462785506, 4.90651111433906934923061444325, 5.50859115809739797985054492547, 6.31268869311300036962581402023, 6.88954559492186336378664122714, 7.72184729672029655294569433513

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