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Properties

Label 2-399-133.40-c1-0-25
Degree $2$
Conductor $399$
Sign $-0.978 + 0.205i$
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  + (1.62 − 1.93i)2-s + (0.173 − 0.984i)3-s + (−0.759 − 4.30i)4-s + (−1.95 − 0.344i)5-s + (−1.62 − 1.93i)6-s + (2.62 − 0.330i)7-s + (−5.18 − 2.99i)8-s + (−0.939 − 0.342i)9-s + (−3.84 + 3.22i)10-s + 0.413·11-s − 4.37·12-s + (−2.21 + 1.85i)13-s + (3.62 − 5.61i)14-s + (−0.679 + 1.86i)15-s + (−5.99 + 2.18i)16-s + (0.435 + 1.19i)17-s + ⋯
L(s)  = 1  + (1.14 − 1.36i)2-s + (0.100 − 0.568i)3-s + (−0.379 − 2.15i)4-s + (−0.874 − 0.154i)5-s + (−0.662 − 0.789i)6-s + (0.992 − 0.124i)7-s + (−1.83 − 1.05i)8-s + (−0.313 − 0.114i)9-s + (−1.21 + 1.01i)10-s + 0.124·11-s − 1.26·12-s + (−0.613 + 0.514i)13-s + (0.967 − 1.49i)14-s + (−0.175 + 0.481i)15-s + (−1.49 + 0.545i)16-s + (0.105 + 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233838 - 2.24712i\)
\(L(\frac12)\) \(\approx\) \(0.233838 - 2.24712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-2.62 + 0.330i)T \)
19 \( 1 + (-4.17 - 1.25i)T \)
good2 \( 1 + (-1.62 + 1.93i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.95 + 0.344i)T + (4.69 + 1.71i)T^{2} \)
11 \( 1 - 0.413T + 11T^{2} \)
13 \( 1 + (2.21 - 1.85i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.435 - 1.19i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-4.69 + 3.94i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.49 - 0.440i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.18 - 2.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 - 2.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.86 + 4.91i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-9.48 + 3.45i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.43 - 6.69i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.95 - 0.874i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-8.94 + 3.25i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-9.02 - 10.7i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.78 - 3.31i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.75 + 13.0i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.81 + 1.55i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-4.15 - 11.4i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.3 - 7.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.587 - 3.33i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.230 - 1.30i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.30044255199921376162932719915, −10.41836135937430367602695440653, −9.213881452889969302797608303926, −8.048147928401500967749485569458, −7.05985440598412088107605852444, −5.56364547809082240564707590176, −4.64209809715263011737345409276, −3.76635610751951446792094620602, −2.47026070743060438052848674391, −1.18102324950068681793605968537, 3.07853947977102294540751674084, 4.09558544972883932965674105990, 5.00394389119752891921685198008, 5.64566108248060669976175105884, 7.19283672114685664584632811085, 7.65744723146491974635377014017, 8.504892003702399835189145594842, 9.682362449164306787705575797850, 11.27689337989886563484827393198, 11.67746837982209551673904584915

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