↑

Properties

Label 2-2667-2667.1235-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.286 + 0.958i$
Analytic cond. $1.33100$
Root an. cond. $1.15369$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.661 − 0.749i)3-s + (0.0747 − 0.997i)4-s + (−0.173 + 0.984i)7-s + (−0.124 + 0.992i)9-s + (−0.797 + 0.603i)12-s + (1.21 + 0.875i)13-s + (−0.988 − 0.149i)16-s − 1.68i·19-s + (0.853 − 0.521i)21-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)27-s + (0.969 + 0.246i)28-s + (1.92 − 0.541i)31-s + (0.980 + 0.198i)36-s + (0.202 + 1.14i)37-s + ⋯
L(s)  = 1  + (−0.661 − 0.749i)3-s + (0.0747 − 0.997i)4-s + (−0.173 + 0.984i)7-s + (−0.124 + 0.992i)9-s + (−0.797 + 0.603i)12-s + (1.21 + 0.875i)13-s + (−0.988 − 0.149i)16-s − 1.68i·19-s + (0.853 − 0.521i)21-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)27-s + (0.969 + 0.246i)28-s + (1.92 − 0.541i)31-s + (0.980 + 0.198i)36-s + (0.202 + 1.14i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9942869749\)
\(L(\frac12)\) \(\approx\) \(0.9942869749\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.661 + 0.749i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
127 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.0747 + 0.997i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.411 + 0.911i)T^{2} \)
13 \( 1 + (-1.21 - 0.875i)T + (0.318 + 0.947i)T^{2} \)
17 \( 1 + (-0.853 - 0.521i)T^{2} \)
19 \( 1 + 1.68iT - T^{2} \)
23 \( 1 + (-0.411 + 0.911i)T^{2} \)
29 \( 1 + (0.542 + 0.840i)T^{2} \)
31 \( 1 + (-1.92 + 0.541i)T + (0.853 - 0.521i)T^{2} \)
37 \( 1 + (-0.202 - 1.14i)T + (-0.939 + 0.342i)T^{2} \)
41 \( 1 + (-0.0249 + 0.999i)T^{2} \)
43 \( 1 + (0.307 + 0.503i)T + (-0.456 + 0.889i)T^{2} \)
47 \( 1 + (-0.988 - 0.149i)T^{2} \)
53 \( 1 + (0.270 + 0.962i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-1.33 + 0.523i)T + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (1.21 + 0.409i)T + (0.797 + 0.603i)T^{2} \)
71 \( 1 + (0.583 + 0.811i)T^{2} \)
73 \( 1 + (-0.228 - 0.742i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (-0.857 + 1.67i)T + (-0.583 - 0.811i)T^{2} \)
83 \( 1 + (-0.698 + 0.715i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.0182 + 0.730i)T + (-0.998 - 0.0498i)T^{2} \)
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

−8.787268417118023077255401171641, −8.299062441055066963168335136611, −7.00295815788408665676252339543, −6.36962073650877484424428538089, −6.10171254145404772640705915912, −5.05074800385461148187119252300, −4.51115465544221932128250315014, −2.80319915600036923902501298929, −1.99551126194708445079562643716, −0.871854069239588450194557514884, 1.15101658537747722884770721666, 3.02453839329939263731816762113, 3.70781057905059599835058630994, 4.20026837889377443524091408483, 5.27575274767949634326722517647, 6.16156832729885569265741478042, 6.81947198600424441428462209892, 7.80507722981455823623210617039, 8.322886907801200565226204154802, 9.205832923272540306384541024502

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