Properties

Label 2-212160-1.1-c1-0-96
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 4·11-s − 13-s − 15-s + 17-s + 8·19-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·33-s + 6·37-s + 39-s + 6·41-s − 4·43-s + 45-s + 4·47-s − 7·49-s − 51-s + 10·53-s − 4·55-s − 8·57-s − 6·61-s − 65-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s − 49-s − 0.140·51-s + 1.37·53-s − 0.539·55-s − 1.05·57-s − 0.768·61-s − 0.124·65-s + 0.963·69-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: $\chi_{212160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 212160,\ (\ :1/2),\ -1)\)

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

−13.19708382752526, −12.82852078190431, −12.27445445474322, −11.68464248322276, −11.59953681984438, −10.80036898292212, −10.45887121239075, −9.958273582833800, −9.579383394495299, −9.235989254149079, −8.425541906764383, −7.846025009676550, −7.533414152461404, −7.206990876646244, −6.316265019737871, −5.940537006863478, −5.528448231453618, −5.082677028333251, −4.610132461158393, −3.844957345470398, −3.369214454807361, −2.608376617000022, −2.191746899090918, −1.450978609786027, −0.7416042806773189, 0, 0.7416042806773189, 1.450978609786027, 2.191746899090918, 2.608376617000022, 3.369214454807361, 3.844957345470398, 4.610132461158393, 5.082677028333251, 5.528448231453618, 5.940537006863478, 6.316265019737871, 7.206990876646244, 7.533414152461404, 7.846025009676550, 8.425541906764383, 9.235989254149079, 9.579383394495299, 9.958273582833800, 10.45887121239075, 10.80036898292212, 11.59953681984438, 11.68464248322276, 12.27445445474322, 12.82852078190431, 13.19708382752526

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