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Properties

Label 2-399-399.32-c1-0-10
Degree $2$
Conductor $399$
Sign $-0.0664 - 0.997i$
Analytic cond. $3.18603$
Root an. cond. $1.78494$
Motivic weight $1$
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 + 1.32i)3-s + (−1.53 − 1.28i)4-s + (−1.28 + 2.31i)7-s + (−0.520 + 2.95i)9-s − 3.46i·12-s + (1.99 + 5.47i)13-s + (0.694 + 3.93i)16-s + (0.5 + 4.33i)19-s + (−4.5 + 0.866i)21-s + (0.868 − 4.92i)25-s + (−4.5 + 2.59i)27-s + (4.94 − 1.88i)28-s + (7.27 − 4.19i)31-s + (4.59 − 3.85i)36-s + (−9.57 + 5.52i)37-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (−0.766 − 0.642i)4-s + (−0.486 + 0.873i)7-s + (−0.173 + 0.984i)9-s − 0.999i·12-s + (0.552 + 1.51i)13-s + (0.173 + 0.984i)16-s + (0.114 + 0.993i)19-s + (−0.981 + 0.188i)21-s + (0.173 − 0.984i)25-s + (−0.866 + 0.499i)27-s + (0.934 − 0.356i)28-s + (1.30 − 0.754i)31-s + (0.766 − 0.642i)36-s + (−1.57 + 0.908i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0664 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0664 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.0664 - 0.997i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ -0.0664 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834085 + 0.891504i\)
\(L(\frac12)\) \(\approx\) \(0.834085 + 0.891504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (1.28 - 2.31i)T \)
19 \( 1 + (-0.5 - 4.33i)T \)
good2 \( 1 + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.868 + 4.92i)T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1.99 - 5.47i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-7.27 + 4.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.57 - 5.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.53 + 8.71i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.69 + 3.16i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-5.59 - 15.3i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-3.56 + 2.99i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-10.5 + 1.85i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (12.2 + 14.5i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.43356267171293063388528053240, −10.18866527095193850040286017949, −9.756915913129712218808051125488, −8.756644892889158005576398833136, −8.397533871313785600311344162710, −6.61474183158934496561877301055, −5.61583700484902542248065144480, −4.55293315101465573128546993198, −3.62044400889006949829225446470, −2.04683656357540774098557339537, 0.789694901505097848484307887486, 2.99474643768330642808540544226, 3.67222805933512045424176546899, 5.11334658838148171892868099850, 6.57217401880457826366502339426, 7.47489321343720589550504010633, 8.203177756511044583662574789515, 9.038679282707828397527539527122, 9.962494588219639292983525195285, 11.04231908870927944800380434910

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