Properties

Label 2-43758-1.1-c1-0-6
Degree $2$
Conductor $43758$
Sign $1$
Analytic cond. $349.409$
Root an. cond. $18.6924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 11-s + 13-s − 2·14-s + 16-s + 17-s − 4·19-s + 2·20-s + 22-s + 4·23-s − 25-s + 26-s − 2·28-s + 4·29-s − 6·31-s + 32-s + 34-s − 4·35-s − 2·37-s − 4·38-s + 2·40-s − 12·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.742·29-s − 1.07·31-s + 0.176·32-s + 0.171·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43758\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(349.409\)
Root analytic conductor: \(18.6924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{43758} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43758,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.139539416\)
\(L(\frac12)\) \(\approx\) \(4.139539416\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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

−14.55810377067033, −14.08291818740882, −13.62068942643495, −13.09046848418955, −12.87850387516105, −12.03097539479195, −11.88625817978291, −10.88003777101016, −10.58703087572959, −10.07028133697878, −9.419001248211009, −8.971963895927298, −8.400511057768874, −7.607693326994042, −6.837514999999137, −6.633587494135278, −5.915300123587524, −5.548693448737854, −4.859677474952782, −4.177997972320920, −3.483139515372617, −3.014471446435799, −2.153638806242413, −1.668867880791157, −0.6294567253515093, 0.6294567253515093, 1.668867880791157, 2.153638806242413, 3.014471446435799, 3.483139515372617, 4.177997972320920, 4.859677474952782, 5.548693448737854, 5.915300123587524, 6.633587494135278, 6.837514999999137, 7.607693326994042, 8.400511057768874, 8.971963895927298, 9.419001248211009, 10.07028133697878, 10.58703087572959, 10.88003777101016, 11.88625817978291, 12.03097539479195, 12.87850387516105, 13.09046848418955, 13.62068942643495, 14.08291818740882, 14.55810377067033

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