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Properties

Label 2-86490-1.1-c1-0-24
Degree $2$
Conductor $86490$
Sign $-1$
Analytic cond. $690.626$
Root an. cond. $26.2797$
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  − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s − 6·13-s + 16-s + 2·17-s + 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s + 6·26-s + 6·29-s − 32-s − 2·34-s + 2·37-s − 4·38-s + 40-s − 10·41-s + 4·43-s − 4·44-s + 8·46-s − 7·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 1.17·26-s + 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s − 49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(690.626\)
Root analytic conductor: \(26.2797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 86490,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
31 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.25220382225353, −13.77985621419450, −12.99780579326141, −12.58973361618100, −12.02966393324171, −11.75345390216189, −11.24148578739344, −10.41512926367863, −10.08948332079017, −9.866813477096875, −9.246913572310341, −8.464909427321289, −8.001945541018750, −7.732761721307624, −7.230610239302734, −6.658736369894818, −5.975882497167567, −5.296359033960444, −4.925459119880938, −4.282502412442245, −3.393925972599432, −2.897004513543077, −2.322447430666720, −1.661314229439746, −0.6368799912874727, 0, 0.6368799912874727, 1.661314229439746, 2.322447430666720, 2.897004513543077, 3.393925972599432, 4.282502412442245, 4.925459119880938, 5.296359033960444, 5.975882497167567, 6.658736369894818, 7.230610239302734, 7.732761721307624, 8.001945541018750, 8.464909427321289, 9.246913572310341, 9.866813477096875, 10.08948332079017, 10.41512926367863, 11.24148578739344, 11.75345390216189, 12.02966393324171, 12.58973361618100, 12.99780579326141, 13.77985621419450, 14.25220382225353

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