↑

Properties

Label 2-2340-13.9-c1-0-13
Degree $2$
Conductor $2340$
Sign $0.872 + 0.488i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + (0.5 + 0.866i)7-s + (1.5 − 2.59i)11-s + (1 + 3.46i)13-s + (−1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s + (4.5 − 7.79i)23-s + 25-s + (−4.5 + 7.79i)29-s + 8·31-s + (0.5 + 0.866i)35-s + (3.5 − 6.06i)37-s + (1.5 − 2.59i)41-s + (0.5 + 0.866i)43-s + (3 − 5.19i)49-s + ⋯
L(s)  = 1  + 0.447·5-s + (0.188 + 0.327i)7-s + (0.452 − 0.783i)11-s + (0.277 + 0.960i)13-s + (−0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s + (0.938 − 1.62i)23-s + 0.200·25-s + (−0.835 + 1.44i)29-s + 1.43·31-s + (0.0845 + 0.146i)35-s + (0.575 − 0.996i)37-s + (0.234 − 0.405i)41-s + (0.0762 + 0.132i)43-s + (0.428 − 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.037034048\)
\(L(\frac12)\) \(\approx\) \(2.037034048\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + (-1 - 3.46i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 7.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.815663704734760938512032144373, −8.553429383932978960344497285537, −7.13477955576465337576446470334, −6.66370491211076931508791306262, −5.86751479854887688953622662848, −4.90342054065998181266788109978, −4.21061932549473637190861834557, −2.96847211873991689843911186418, −2.15655547608844431942471640989, −0.811446194337417660079124394338, 1.15075681083104921035062135474, 2.13129367445823359700598163322, 3.35175041987842435503183080552, 4.21352412332947996950622474244, 5.09571113841103880455206903261, 6.04086070554811934187130170812, 6.57257019774244667581609677797, 7.77099735324449532980487119907, 8.031054992288858227986354698404, 9.211966508324690687834619285038

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