Properties

Label 2-164730-1.1-c1-0-47
Degree $2$
Conductor $164730$
Sign $-1$
Analytic cond. $1315.37$
Root an. cond. $36.2681$
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 + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s − 4·11-s + 12-s + 2·14-s + 15-s + 16-s − 18-s − 19-s + 20-s − 2·21-s + 4·22-s − 4·23-s − 24-s + 25-s + 27-s − 2·28-s + 8·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.436·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.48·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164730\)    =    \(2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(1315.37\)
Root analytic conductor: \(36.2681\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 164730,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
17 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.52361553405514, −13.03192324395014, −12.56877569595037, −12.06905893933093, −11.66562764538532, −10.86950650153435, −10.35937409132807, −10.03702183711699, −9.918885923612308, −9.091956489557046, −8.709327887733631, −8.228528947850548, −7.846781532801326, −7.275445638084773, −6.605217706471370, −6.378582457362063, −5.742787448783600, −5.059003558896320, −4.610404757240782, −3.781487578896653, −3.195323466497640, −2.659393085034975, −2.301329756607665, −1.564843637587549, −0.7647761341836346, 0, 0.7647761341836346, 1.564843637587549, 2.301329756607665, 2.659393085034975, 3.195323466497640, 3.781487578896653, 4.610404757240782, 5.059003558896320, 5.742787448783600, 6.378582457362063, 6.605217706471370, 7.275445638084773, 7.846781532801326, 8.228528947850548, 8.709327887733631, 9.091956489557046, 9.918885923612308, 10.03702183711699, 10.35937409132807, 10.86950650153435, 11.66562764538532, 12.06905893933093, 12.56877569595037, 13.03192324395014, 13.52361553405514

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