Properties

Label 2-212160-1.1-c1-0-87
Degree $2$
Conductor $212160$
Sign $1$
Analytic cond. $1694.10$
Root an. cond. $41.1595$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 13-s + 15-s − 17-s − 4·19-s + 4·21-s + 8·23-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·33-s + 4·35-s + 2·37-s − 39-s + 6·41-s − 4·43-s + 45-s + 9·49-s − 51-s + 6·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.140·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212160\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1694.10\)
Root analytic conductor: \(41.1595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 212160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.604268491\)
\(L(\frac12)\) \(\approx\) \(5.604268491\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.02446199283448, −12.65762537488603, −12.13639805652296, −11.51529622611738, −11.06733823752900, −10.62466099037450, −10.32026664897612, −9.744932579948235, −9.129387897047018, −8.679836146405037, −8.213769016740594, −7.948115701222278, −7.426573270421035, −6.778790556552409, −6.389053399690730, −5.627439974482684, −4.985367408472401, −4.819028615581763, −4.377849550993568, −3.530134181349204, −2.839599389091088, −2.321296238688873, −2.094583910757162, −1.126317774610779, −0.6984698860436189, 0.6984698860436189, 1.126317774610779, 2.094583910757162, 2.321296238688873, 2.839599389091088, 3.530134181349204, 4.377849550993568, 4.819028615581763, 4.985367408472401, 5.627439974482684, 6.389053399690730, 6.778790556552409, 7.426573270421035, 7.948115701222278, 8.213769016740594, 8.679836146405037, 9.129387897047018, 9.744932579948235, 10.32026664897612, 10.62466099037450, 11.06733823752900, 11.51529622611738, 12.13639805652296, 12.65762537488603, 13.02446199283448

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