Properties

Label 2-14586-1.1-c1-0-9
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 8·19-s + 20-s + 21-s + 22-s − 7·23-s − 24-s − 4·25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-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 \( 1 - T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.36755248150315, −15.78876512266161, −15.41166535324832, −14.74503269123112, −14.07977442555934, −13.62200572988908, −13.24952898803623, −12.31035120872395, −11.75365355234884, −11.31023315784556, −10.52235956775744, −9.940406440056929, −9.486235530679970, −9.066869277418127, −8.150650812844574, −7.737278254362530, −7.441921556433914, −6.233578636816883, −6.025069482711994, −5.060826783636213, −4.337663832507605, −3.379944124673377, −2.810164075493390, −1.835127771410169, −1.379088870055644, 0, 1.379088870055644, 1.835127771410169, 2.810164075493390, 3.379944124673377, 4.337663832507605, 5.060826783636213, 6.025069482711994, 6.233578636816883, 7.441921556433914, 7.737278254362530, 8.150650812844574, 9.066869277418127, 9.486235530679970, 9.940406440056929, 10.52235956775744, 11.31023315784556, 11.75365355234884, 12.31035120872395, 13.24952898803623, 13.62200572988908, 14.07977442555934, 14.74503269123112, 15.41166535324832, 15.78876512266161, 16.36755248150315

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