↑

Properties

Label 2-299-299.114-c0-0-0
Degree $2$
Conductor $299$
Sign $-0.632 - 0.774i$
Analytic cond. $0.149220$
Root an. cond. $0.386290$
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  + (−1.70 + 0.984i)2-s + (0.766 + 1.32i)3-s + (1.43 − 2.49i)4-s + (−2.61 − 1.50i)6-s + 3.70i·8-s + (−0.673 + 1.16i)9-s + 4.41·12-s + (0.173 + 0.984i)13-s + (−2.20 − 3.82i)16-s − 2.65i·18-s + (−0.5 − 0.866i)23-s + (−4.91 + 2.83i)24-s − 25-s + (−1.26 − 1.50i)26-s − 0.532·27-s + ⋯
L(s)  = 1  + (−1.70 + 0.984i)2-s + (0.766 + 1.32i)3-s + (1.43 − 2.49i)4-s + (−2.61 − 1.50i)6-s + 3.70i·8-s + (−0.673 + 1.16i)9-s + 4.41·12-s + (0.173 + 0.984i)13-s + (−2.20 − 3.82i)16-s − 2.65i·18-s + (−0.5 − 0.866i)23-s + (−4.91 + 2.83i)24-s − 25-s + (−1.26 − 1.50i)26-s − 0.532·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(0.149220\)
Root analytic conductor: \(0.386290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :0),\ -0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4622020429\)
\(L(\frac12)\) \(\approx\) \(0.4622020429\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.28iT - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 1.96iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 0.684iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−11.71009586743335766220753430987, −10.77601958421415633983951179152, −10.00316576473833285284997626861, −9.376988389596390930255153362426, −8.684017177835628720111613006661, −7.906995262336724007694139186868, −6.74531088779858849272441276528, −5.60834362079694936214853210543, −4.24342911037498609944567819001, −2.25416813820693029840901358088, 1.30914334637195636000935397625, 2.47973191849619736266050775398, 3.49792566224398728540772887025, 6.32815259831209767459582453820, 7.48682021844471393231625504715, 7.908919664605635574412183455579, 8.729462023342472008889165958396, 9.644393595979626285841730498813, 10.58935192635467786358049941092, 11.61022307689937341580642210282

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