Properties

Label 2-212160-1.1-c1-0-30
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 − 2·7-s + 9-s − 13-s + 15-s − 17-s + 2·19-s − 2·21-s + 25-s + 27-s + 6·29-s + 4·31-s − 2·35-s + 10·37-s − 39-s − 6·41-s − 10·43-s + 45-s − 3·49-s − 51-s + 6·53-s + 2·57-s − 6·59-s − 14·61-s − 2·63-s − 65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.64·37-s − 0.160·39-s − 0.937·41-s − 1.52·43-s + 0.149·45-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.264·57-s − 0.781·59-s − 1.79·61-s − 0.251·63-s − 0.124·65-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: \(0\)
Selberg data: \((2,\ 212160,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.045055609\)
\(L(\frac12)\) \(\approx\) \(3.045055609\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−13.20756319845058, −12.58674458636042, −12.01294116282588, −11.87083744955640, −10.97461479344088, −10.66639002559104, −9.982681303546423, −9.691379679789250, −9.412589055070616, −8.788571885390941, −8.198351560088098, −7.956249157207384, −7.245471088573706, −6.649049983366146, −6.426265121546084, −5.891960643602001, −4.997880316247903, −4.876285565771924, −4.072669653694584, −3.508900897774582, −2.928152518127058, −2.597137793186307, −1.887255675438033, −1.205843444075903, −0.4821740485309796, 0.4821740485309796, 1.205843444075903, 1.887255675438033, 2.597137793186307, 2.928152518127058, 3.508900897774582, 4.072669653694584, 4.876285565771924, 4.997880316247903, 5.891960643602001, 6.426265121546084, 6.649049983366146, 7.245471088573706, 7.956249157207384, 8.198351560088098, 8.788571885390941, 9.412589055070616, 9.691379679789250, 9.982681303546423, 10.66639002559104, 10.97461479344088, 11.87083744955640, 12.01294116282588, 12.58674458636042, 13.20756319845058

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