Properties

Label 2-206400-1.1-c1-0-131
Degree $2$
Conductor $206400$
Sign $1$
Analytic cond. $1648.11$
Root an. cond. $40.5969$
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 + 2·7-s + 9-s + 6·11-s + 2·13-s − 2·17-s + 4·19-s − 2·21-s − 27-s + 8·29-s − 6·33-s − 2·37-s − 2·39-s − 6·41-s + 43-s + 4·47-s − 3·49-s + 2·51-s + 2·53-s − 4·57-s + 6·59-s + 2·61-s + 2·63-s − 8·67-s − 12·71-s + 8·73-s + 12·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.436·21-s − 0.192·27-s + 1.48·29-s − 1.04·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s + 0.583·47-s − 3/7·49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s + 0.781·59-s + 0.256·61-s + 0.251·63-s − 0.977·67-s − 1.42·71-s + 0.936·73-s + 1.36·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(206400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(1648.11\)
Root analytic conductor: \(40.5969\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 206400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.751321436\)
\(L(\frac12)\) \(\approx\) \(3.751321436\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.04895479483230, −12.31927225090358, −11.98616381903757, −11.69200680473067, −11.24644459644678, −10.84520617881255, −10.18875646600030, −9.863615771252259, −9.157544728301012, −8.791284180047963, −8.427834532092725, −7.742417460235399, −7.194187050674144, −6.737494146938917, −6.279609750063270, −5.888918798552288, −5.171310041901195, −4.674060604226467, −4.333716662359845, −3.572105729503886, −3.286324569394510, −2.254606840208753, −1.690580235649550, −1.094114923375179, −0.6623880434512171, 0.6623880434512171, 1.094114923375179, 1.690580235649550, 2.254606840208753, 3.286324569394510, 3.572105729503886, 4.333716662359845, 4.674060604226467, 5.171310041901195, 5.888918798552288, 6.279609750063270, 6.737494146938917, 7.194187050674144, 7.742417460235399, 8.427834532092725, 8.791284180047963, 9.157544728301012, 9.863615771252259, 10.18875646600030, 10.84520617881255, 11.24644459644678, 11.69200680473067, 11.98616381903757, 12.31927225090358, 13.04895479483230

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