Properties

Label 2-212160-1.1-c1-0-141
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 + 2·11-s + 13-s − 15-s − 17-s − 6·19-s + 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s + 2·31-s + 2·33-s − 2·35-s + 2·37-s + 39-s − 10·41-s + 8·43-s − 45-s + 6·47-s − 3·49-s − 51-s + 2·53-s − 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s − 0.338·35-s + 0.328·37-s + 0.160·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s − 0.269·55-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
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

−13.14518536121373, −12.89392153528843, −12.23781146723761, −11.86347101713932, −11.35024298316057, −11.01413973957315, −10.28032529313752, −10.17214410382250, −9.283525041471374, −8.948031747637845, −8.448836197110320, −8.156407713558244, −7.592218849122189, −7.157776660024074, −6.455396362812029, −6.204415954278856, −5.442515911095962, −4.808489027103661, −4.346469068981536, −3.856564896047671, −3.527342252183851, −2.593579840739012, −2.163380288189031, −1.589281230825229, −0.8892218546756005, 0, 0.8892218546756005, 1.589281230825229, 2.163380288189031, 2.593579840739012, 3.527342252183851, 3.856564896047671, 4.346469068981536, 4.808489027103661, 5.442515911095962, 6.204415954278856, 6.455396362812029, 7.157776660024074, 7.592218849122189, 8.156407713558244, 8.448836197110320, 8.948031747637845, 9.283525041471374, 10.17214410382250, 10.28032529313752, 11.01413973957315, 11.35024298316057, 11.86347101713932, 12.23781146723761, 12.89392153528843, 13.14518536121373

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