Properties

Label 2-212160-1.1-c1-0-113
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 − 2·7-s + 9-s + 3·11-s − 13-s + 15-s − 17-s + 7·19-s + 2·21-s + 4·23-s + 25-s − 27-s + 4·29-s − 11·31-s − 3·33-s + 2·35-s + 9·37-s + 39-s + 4·41-s + 11·43-s − 45-s − 6·47-s − 3·49-s + 51-s − 6·53-s − 3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 1.60·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.97·31-s − 0.522·33-s + 0.338·35-s + 1.47·37-s + 0.160·39-s + 0.624·41-s + 1.67·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.404·55-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.97652082929700, −12.77803068532757, −12.30894953807878, −11.84936598676423, −11.32034056452673, −11.02028852225198, −10.62371371019725, −9.699593218756944, −9.560591443800734, −9.201863920047116, −8.628793860475056, −7.769396431584436, −7.522269121688411, −7.035004541220423, −6.495840811317039, −6.060554691848634, −5.560686252389375, −4.931645235544507, −4.470833244708882, −3.900022720077725, −3.272236250497722, −2.964664007535975, −2.098003018238758, −1.255374750478727, −0.8091701366577640, 0, 0.8091701366577640, 1.255374750478727, 2.098003018238758, 2.964664007535975, 3.272236250497722, 3.900022720077725, 4.470833244708882, 4.931645235544507, 5.560686252389375, 6.060554691848634, 6.495840811317039, 7.035004541220423, 7.522269121688411, 7.769396431584436, 8.628793860475056, 9.201863920047116, 9.560591443800734, 9.699593218756944, 10.62371371019725, 11.02028852225198, 11.32034056452673, 11.84936598676423, 12.30894953807878, 12.77803068532757, 12.97652082929700

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