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Properties

Label 2-299-299.114-c0-0-2
Degree $2$
Conductor $299$
Sign $-0.354 + 0.935i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 − 0.642i)2-s + (−0.939 − 1.62i)3-s + (0.326 − 0.565i)4-s + (−2.09 − 1.20i)6-s + 0.446i·8-s + (−1.26 + 2.19i)9-s − 1.22·12-s + (0.766 − 0.642i)13-s + (0.613 + 1.06i)16-s + 3.25i·18-s + (−0.5 − 0.866i)23-s + (0.726 − 0.419i)24-s − 25-s + (0.439 − 1.20i)26-s + 2.87·27-s + ⋯
L(s)  = 1  + (1.11 − 0.642i)2-s + (−0.939 − 1.62i)3-s + (0.326 − 0.565i)4-s + (−2.09 − 1.20i)6-s + 0.446i·8-s + (−1.26 + 2.19i)9-s − 1.22·12-s + (0.766 − 0.642i)13-s + (0.613 + 1.06i)16-s + 3.25i·18-s + (−0.5 − 0.866i)23-s + (0.726 − 0.419i)24-s − 25-s + (0.439 − 1.20i)26-s + 2.87·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $-0.354 + 0.935i$
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: \((2,\ 299,\ (\ :0),\ -0.354 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9812385746\)
\(L(\frac12)\) \(\approx\) \(0.9812385746\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 0.684iT - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.28iT - T^{2} \)
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 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.96iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03512760371743774721896612787, −11.18044859122400418974273758966, −10.48690515744390345163294731908, −8.478351586097661504299296941255, −7.68975522676111160120854813263, −6.39017644833699576671827921051, −5.76807462483322463207632117535, −4.67214313732893305457178922367, −2.99982777685841159506363002746, −1.64105632702645315374959372320, 3.61410886568110032811303808108, 4.20291965281873156129532022129, 5.27055425394881661938174309464, 5.91968013811819827111234448556, 6.85216402492089177463203157871, 8.607594410143464037522357627997, 9.767402687817733460795105235893, 10.31171405178939903539515168365, 11.59440591736473515990798856692, 11.96699490037047451053946586175

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