Properties

Label 2-164730-1.1-c1-0-35
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 − 4·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s − 2·13-s − 4·14-s − 15-s + 16-s + 18-s − 19-s + 20-s + 4·21-s − 4·22-s − 4·23-s − 24-s + 25-s − 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.46958620523344, −13.02953813454079, −12.49827744846684, −12.16354574196700, −11.91314917011578, −11.08306056515639, −10.44901757201856, −10.31732947741011, −9.836263495832869, −9.458126228228338, −8.556397817841802, −8.227259283027857, −7.461374987659791, −6.940084178739232, −6.543549977873828, −6.147308367460801, −5.629429558543035, −4.993832336238571, −4.809235439620750, −3.963176435379017, −3.279082485275113, −2.994135031560669, −2.256341731540652, −1.759279613842497, −0.6401485172934032, 0, 0.6401485172934032, 1.759279613842497, 2.256341731540652, 2.994135031560669, 3.279082485275113, 3.963176435379017, 4.809235439620750, 4.993832336238571, 5.629429558543035, 6.147308367460801, 6.543549977873828, 6.940084178739232, 7.461374987659791, 8.227259283027857, 8.556397817841802, 9.458126228228338, 9.836263495832869, 10.31732947741011, 10.44901757201856, 11.08306056515639, 11.91314917011578, 12.16354574196700, 12.49827744846684, 13.02953813454079, 13.46958620523344

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