Properties

Label 2-14586-1.1-c1-0-7
Degree $2$
Conductor $14586$
Sign $-1$
Analytic cond. $116.469$
Root an. cond. $10.7921$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s + 13-s − 2·14-s + 16-s − 17-s − 18-s + 4·19-s − 2·21-s − 22-s + 6·23-s + 24-s − 5·25-s − 26-s − 27-s + 2·28-s − 8·29-s + 2·31-s − 32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s − 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14586\)    =    \(2 \cdot 3 \cdot 11 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(116.469\)
Root analytic conductor: \(10.7921\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14586} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14586,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.53258422360301, −15.93525309148417, −15.24392051134339, −14.96774119063243, −14.20329646795771, −13.47766009736874, −13.07196492046423, −12.16393245686485, −11.69921357781111, −11.26474054893049, −10.80238989660613, −10.14153788904966, −9.377484650574977, −9.108171036168251, −8.179318011591290, −7.742956628661186, −7.081031270327253, −6.516955998966931, −5.675397611648862, −5.244769853821002, −4.404792047978282, −3.636218196978220, −2.740746396114859, −1.699608028230151, −1.174951689611294, 0, 1.174951689611294, 1.699608028230151, 2.740746396114859, 3.636218196978220, 4.404792047978282, 5.244769853821002, 5.675397611648862, 6.516955998966931, 7.081031270327253, 7.742956628661186, 8.179318011591290, 9.108171036168251, 9.377484650574977, 10.14153788904966, 10.80238989660613, 11.26474054893049, 11.69921357781111, 12.16393245686485, 13.07196492046423, 13.47766009736874, 14.20329646795771, 14.96774119063243, 15.24392051134339, 15.93525309148417, 16.53258422360301

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