Properties

Label 2-16830-1.1-c1-0-16
Degree $2$
Conductor $16830$
Sign $-1$
Analytic cond. $134.388$
Root an. cond. $11.5925$
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 − 4·7-s − 8-s + 10-s − 11-s − 4·13-s + 4·14-s + 16-s − 17-s − 2·19-s − 20-s + 22-s − 2·23-s + 25-s + 4·26-s − 4·28-s − 2·29-s − 32-s + 34-s + 4·35-s + 10·37-s + 2·38-s + 40-s − 10·41-s + 2·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s − 0.371·29-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 1.64·37-s + 0.324·38-s + 0.158·40-s − 1.56·41-s + 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(134.388\)
Root analytic conductor: \(11.5925\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16830,\ (\ :1/2),\ -1)\)

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

−16.34278730207397, −15.66883834943621, −15.11263376251177, −14.89755780492055, −13.83102854531416, −13.40274956659460, −12.60042363889376, −12.38167275609501, −11.74089944900311, −11.02475917233029, −10.41361577190852, −9.890220305788905, −9.477723638319432, −8.865403737920613, −8.192879444911340, −7.509940430855706, −7.045360723859248, −6.435050272844027, −5.841507549348028, −5.023219922131779, −4.125687554391892, −3.494623836529722, −2.651171605854609, −2.185597918146964, −0.7481226000441284, 0, 0.7481226000441284, 2.185597918146964, 2.651171605854609, 3.494623836529722, 4.125687554391892, 5.023219922131779, 5.841507549348028, 6.435050272844027, 7.045360723859248, 7.509940430855706, 8.192879444911340, 8.865403737920613, 9.477723638319432, 9.890220305788905, 10.41361577190852, 11.02475917233029, 11.74089944900311, 12.38167275609501, 12.60042363889376, 13.40274956659460, 13.83102854531416, 14.89755780492055, 15.11263376251177, 15.66883834943621, 16.34278730207397

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