Properties

Label 2-212160-1.1-c1-0-133
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 − 4·7-s + 9-s + 4·11-s + 13-s − 15-s − 17-s + 8·19-s − 4·21-s − 8·23-s + 25-s + 27-s + 10·29-s − 8·31-s + 4·33-s + 4·35-s + 6·37-s + 39-s + 2·41-s − 4·43-s − 45-s + 8·47-s + 9·49-s − 51-s − 6·53-s − 4·55-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 + 1.83·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s + 0.986·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.539·55-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.38097303219978, −12.66724248775751, −12.37083036369582, −11.84793048683529, −11.57568754674160, −10.91652592456466, −10.19006833535631, −9.906297477774770, −9.459940338613739, −9.109081217969809, −8.564226681562112, −8.068696345554080, −7.455949876459326, −7.067406138918814, −6.555737650347477, −6.114381622753258, −5.651693384636674, −4.883392026716564, −4.153387763548697, −3.785373531630649, −3.429483513619220, −2.826723406785059, −2.308757210347356, −1.368084597307830, −0.8691474949366849, 0, 0.8691474949366849, 1.368084597307830, 2.308757210347356, 2.826723406785059, 3.429483513619220, 3.785373531630649, 4.153387763548697, 4.883392026716564, 5.651693384636674, 6.114381622753258, 6.555737650347477, 7.067406138918814, 7.455949876459326, 8.068696345554080, 8.564226681562112, 9.109081217969809, 9.459940338613739, 9.906297477774770, 10.19006833535631, 10.91652592456466, 11.57568754674160, 11.84793048683529, 12.37083036369582, 12.66724248775751, 13.38097303219978

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