↑

Properties

Label 2-299-299.160-c0-0-2
Degree $2$
Conductor $299$
Sign $0.986 + 0.161i$
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  + (0.592 + 0.342i)2-s + (0.173 − 0.300i)3-s + (−0.266 − 0.460i)4-s + (0.205 − 0.118i)6-s − 1.04i·8-s + (0.439 + 0.761i)9-s − 0.184·12-s + (−0.939 + 0.342i)13-s + (0.0923 − 0.160i)16-s + 0.601i·18-s + (−0.5 + 0.866i)23-s + (−0.315 − 0.181i)24-s − 25-s + (−0.673 − 0.118i)26-s + 0.652·27-s + ⋯
L(s)  = 1  + (0.592 + 0.342i)2-s + (0.173 − 0.300i)3-s + (−0.266 − 0.460i)4-s + (0.205 − 0.118i)6-s − 1.04i·8-s + (0.439 + 0.761i)9-s − 0.184·12-s + (−0.939 + 0.342i)13-s + (0.0923 − 0.160i)16-s + 0.601i·18-s + (−0.5 + 0.866i)23-s + (−0.315 − 0.181i)24-s − 25-s + (−0.673 − 0.118i)26-s + 0.652·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $0.986 + 0.161i$
Analytic conductor: \(0.149220\)
Root analytic conductor: \(0.386290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :0),\ 0.986 + 0.161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9871135091\)
\(L(\frac12)\) \(\approx\) \(0.9871135091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.96iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + 0.684iT - T^{2} \)
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 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.28iT - T^{2} \)
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

−12.17891591329931856953286467266, −11.03821271986473995448202056910, −9.948815365122484200214181432151, −9.309582838256793330123675891233, −7.82210802214272545691957053918, −7.09912143596171419732108281252, −5.86743904560511903108317603095, −4.94945502179875678007590050744, −3.87848955710667483028000241274, −1.98267904604144642443036249991, 2.50906662764242310474340886110, 3.76385080488174815014788069940, 4.58668668089972911401777969333, 5.83182621001742307512254369100, 7.23761965239575648186865539902, 8.218847091776160692348057269815, 9.272772449017236351439725116081, 10.09961134266964744129842988667, 11.26212418760427335725275094233, 12.33278710398773459001519209529

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