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Properties

Label 2-399-399.263-c0-0-0
Degree $2$
Conductor $399$
Sign $-0.0482 + 0.998i$
Analytic cond. $0.199126$
Root an. cond. $0.446236$
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  + (0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)12-s + (−1.43 − 0.524i)13-s + (0.766 + 0.642i)16-s + 19-s + (−0.5 − 0.866i)21-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.939 + 0.342i)28-s + 0.347·31-s + (0.766 + 0.642i)36-s + (0.939 + 1.62i)37-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)12-s + (−1.43 − 0.524i)13-s + (0.766 + 0.642i)16-s + 19-s + (−0.5 − 0.866i)21-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)27-s + (−0.939 + 0.342i)28-s + 0.347·31-s + (0.766 + 0.642i)36-s + (0.939 + 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.0482 + 0.998i$
Analytic conductor: \(0.199126\)
Root analytic conductor: \(0.446236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :0),\ -0.0482 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7256875578\)
\(L(\frac12)\) \(\approx\) \(0.7256875578\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
37 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−11.39971121350964962093197447840, −10.22767529525538512009781714566, −9.464273766673610219497398723590, −8.236744825370847393035994705925, −7.75615800283399070077671007313, −6.65714965002717709821980833927, −5.34682232659847370524693912663, −4.54587117556312544680734145058, −2.89670239068255967765801712715, −1.14321656026998932375487104648, 2.58586294249349647645969974647, 3.92077648173447361440386377295, 4.94913824892147118531454796889, 5.44230536664703128485029386056, 7.34591780398307979451048884502, 8.271491775377567494938853959393, 9.202772814669257768945021063750, 9.602873384658710745451509750324, 10.74691189946811093514962714645, 11.77109955343334553069363208068

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