Properties

Label 2-6160-1.1-c1-0-60
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  − 2.65·3-s − 5-s + 7-s + 4.05·9-s + 11-s − 2.65·13-s + 2.65·15-s − 5.44·17-s + 4.10·19-s − 2.65·21-s + 0.259·23-s + 25-s − 2.79·27-s − 2·29-s + 4.91·31-s − 2.65·33-s − 35-s − 2.25·37-s + 7.05·39-s − 1.08·41-s + 4.53·43-s − 4.05·45-s − 4.13·47-s + 49-s + 14.4·51-s − 4.36·53-s − 55-s + ⋯
L(s)  = 1  − 1.53·3-s − 0.447·5-s + 0.377·7-s + 1.35·9-s + 0.301·11-s − 0.736·13-s + 0.685·15-s − 1.32·17-s + 0.941·19-s − 0.579·21-s + 0.0541·23-s + 0.200·25-s − 0.537·27-s − 0.371·29-s + 0.882·31-s − 0.462·33-s − 0.169·35-s − 0.371·37-s + 1.12·39-s − 0.169·41-s + 0.691·43-s − 0.603·45-s − 0.603·47-s + 0.142·49-s + 2.02·51-s − 0.599·53-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 \( 1 + 2.65T + 3T^{2} \)
13 \( 1 + 2.65T + 13T^{2} \)
17 \( 1 + 5.44T + 17T^{2} \)
19 \( 1 - 4.10T + 19T^{2} \)
23 \( 1 - 0.259T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4.91T + 31T^{2} \)
37 \( 1 + 2.25T + 37T^{2} \)
41 \( 1 + 1.08T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 4.13T + 47T^{2} \)
53 \( 1 + 4.36T + 53T^{2} \)
59 \( 1 + 5.97T + 59T^{2} \)
61 \( 1 - 3.80T + 61T^{2} \)
67 \( 1 + 7.15T + 67T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + 6.79T + 83T^{2} \)
89 \( 1 - 5.31T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.49379789510451122581915965355, −6.90439910574245381054672512754, −6.29481968872842863479634363187, −5.52963934089536074915408746457, −4.77803431737844659170523619284, −4.44434050789582462950446322638, −3.33282062092290517319335201682, −2.15206026523181725862382305778, −1.00993013567478273050309231425, 0, 1.00993013567478273050309231425, 2.15206026523181725862382305778, 3.33282062092290517319335201682, 4.44434050789582462950446322638, 4.77803431737844659170523619284, 5.52963934089536074915408746457, 6.29481968872842863479634363187, 6.90439910574245381054672512754, 7.49379789510451122581915965355

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