Properties

Label 2-348726-1.1-c1-0-56
Degree $2$
Conductor $348726$
Sign $-1$
Analytic cond. $2784.59$
Root an. cond. $52.7692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 3·5-s + 6-s − 7-s − 8-s + 9-s − 3·10-s + 4·11-s − 12-s + 3·13-s + 14-s − 3·15-s + 16-s − 4·17-s − 18-s + 3·20-s + 21-s − 4·22-s + 23-s + 24-s + 4·25-s − 3·26-s − 27-s − 28-s − 3·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.20·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.670·20-s + 0.218·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348726\)    =    \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2784.59\)
Root analytic conductor: \(52.7692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{348726} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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

−12.96666080894727, −12.12589473795351, −11.72492149982124, −11.45709679145684, −10.85285604430687, −10.49350242887421, −9.953774209664593, −9.620484946036125, −9.235412044308625, −8.813242561587085, −8.348553745537489, −7.787749045660196, −6.975987556655940, −6.675198921908798, −6.278700701544139, −6.080061471178610, −5.406753615533549, −4.915112970506403, −4.207540201950063, −3.761062424078827, −3.024592563587354, −2.459319917309407, −1.795790914230662, −1.400941192450197, −0.8592982037842858, 0, 0.8592982037842858, 1.400941192450197, 1.795790914230662, 2.459319917309407, 3.024592563587354, 3.761062424078827, 4.207540201950063, 4.915112970506403, 5.406753615533549, 6.080061471178610, 6.278700701544139, 6.675198921908798, 6.975987556655940, 7.787749045660196, 8.348553745537489, 8.813242561587085, 9.235412044308625, 9.620484946036125, 9.953774209664593, 10.49350242887421, 10.85285604430687, 11.45709679145684, 11.72492149982124, 12.12589473795351, 12.96666080894727

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