Properties

Label 2-18515-1.1-c1-0-16
Degree $2$
Conductor $18515$
Sign $-1$
Analytic cond. $147.843$
Root an. cond. $12.1590$
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  + 3-s − 2·4-s + 5-s − 7-s − 2·9-s + 3·11-s − 2·12-s + 5·13-s + 15-s + 4·16-s − 3·17-s − 2·19-s − 2·20-s − 21-s + 25-s − 5·27-s + 2·28-s + 3·29-s − 4·31-s + 3·33-s − 35-s + 4·36-s − 2·37-s + 5·39-s − 12·41-s + 10·43-s − 6·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 1/5·25-s − 0.962·27-s + 0.377·28-s + 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s + 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18515 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18515\)    =    \(5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(147.843\)
Root analytic conductor: \(12.1590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{18515} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 18515,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−15.90589448422367, −15.48939924147563, −14.74798182312992, −14.25722618539881, −13.83473074440309, −13.48076421561697, −12.91767763913842, −12.32780564768895, −11.64769470855588, −10.87621996172989, −10.52221039979505, −9.596200783021881, −9.194889317799429, −8.737697883265387, −8.435717966541179, −7.652073183591930, −6.717704462355851, −6.166779005517228, −5.688458569616073, −4.853913598039859, −4.066485563099640, −3.607186244853865, −2.947103349574134, −1.955482144047320, −1.124430087508032, 0, 1.124430087508032, 1.955482144047320, 2.947103349574134, 3.607186244853865, 4.066485563099640, 4.853913598039859, 5.688458569616073, 6.166779005517228, 6.717704462355851, 7.652073183591930, 8.435717966541179, 8.737697883265387, 9.194889317799429, 9.596200783021881, 10.52221039979505, 10.87621996172989, 11.64769470855588, 12.32780564768895, 12.91767763913842, 13.48076421561697, 13.83473074440309, 14.25722618539881, 14.74798182312992, 15.48939924147563, 15.90589448422367

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