Properties

Label 2-188760-1.1-c1-0-53
Degree $2$
Conductor $188760$
Sign $-1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 4·7-s + 9-s − 13-s + 15-s + 2·17-s − 4·19-s − 4·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 4·35-s + 10·37-s + 39-s + 10·41-s − 4·43-s − 45-s + 9·49-s − 2·51-s + 10·53-s + 4·57-s − 12·59-s + 2·61-s + 4·63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.676·35-s + 1.64·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.280·51-s + 1.37·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.21389891319681, −12.86238570257180, −12.22774274493645, −11.94008994404102, −11.44058314477510, −11.05471622895688, −10.67705056547283, −10.25290255812056, −9.412838958636345, −9.263810738126254, −8.362053018126838, −8.028400404670144, −7.704523468624123, −7.226248158667255, −6.533212565592400, −6.004205094984840, −5.495175227330318, −5.049416965106498, −4.374774712083545, −4.173681396984093, −3.552955941452582, −2.559885404663863, −2.143659607594128, −1.439659823769605, −0.8300720554274437, 0, 0.8300720554274437, 1.439659823769605, 2.143659607594128, 2.559885404663863, 3.552955941452582, 4.173681396984093, 4.374774712083545, 5.049416965106498, 5.495175227330318, 6.004205094984840, 6.533212565592400, 7.226248158667255, 7.704523468624123, 8.028400404670144, 8.362053018126838, 9.263810738126254, 9.412838958636345, 10.25290255812056, 10.67705056547283, 11.05471622895688, 11.44058314477510, 11.94008994404102, 12.22774274493645, 12.86238570257180, 13.21389891319681

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