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Properties

Label 2-248256-1.1-c1-0-50
Degree $2$
Conductor $248256$
Sign $-1$
Analytic cond. $1982.33$
Root an. cond. $44.5234$
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  + 3·5-s − 4·7-s + 5·11-s − 4·13-s + 6·17-s − 7·19-s − 3·23-s + 4·25-s + 3·29-s − 4·31-s − 12·35-s − 8·37-s + 6·41-s − 8·43-s + 6·47-s + 9·49-s + 53-s + 15·55-s + 9·59-s − 2·61-s − 12·65-s + 10·67-s + 8·71-s − 6·73-s − 20·77-s − 16·79-s + 16·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s + 1.50·11-s − 1.10·13-s + 1.45·17-s − 1.60·19-s − 0.625·23-s + 4/5·25-s + 0.557·29-s − 0.718·31-s − 2.02·35-s − 1.31·37-s + 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.137·53-s + 2.02·55-s + 1.17·59-s − 0.256·61-s − 1.48·65-s + 1.22·67-s + 0.949·71-s − 0.702·73-s − 2.27·77-s − 1.80·79-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248256\)    =    \(2^{6} \cdot 3^{2} \cdot 431\)
Sign: $-1$
Analytic conductor: \(1982.33\)
Root analytic conductor: \(44.5234\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 248256,\ (\ :1/2),\ -1)\)

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 \( 1 \)
431 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.00147929422808, −12.59653816310212, −12.24292654593200, −11.99514375783046, −11.21578150562532, −10.53052095896541, −10.13652297538418, −9.817309528657702, −9.562007104406269, −8.969592122711015, −8.668291160758903, −7.936703009894906, −7.194243539177010, −6.835259812725127, −6.332437611238148, −6.147138523679493, −5.439793620905943, −5.154768438642783, −4.165553676222247, −3.868347100967316, −3.255037640176286, −2.635699478920107, −2.095966403566789, −1.590234180029452, −0.7980986914070783, 0, 0.7980986914070783, 1.590234180029452, 2.095966403566789, 2.635699478920107, 3.255037640176286, 3.868347100967316, 4.165553676222247, 5.154768438642783, 5.439793620905943, 6.147138523679493, 6.332437611238148, 6.835259812725127, 7.194243539177010, 7.936703009894906, 8.668291160758903, 8.969592122711015, 9.562007104406269, 9.817309528657702, 10.13652297538418, 10.53052095896541, 11.21578150562532, 11.99514375783046, 12.24292654593200, 12.59653816310212, 13.00147929422808

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