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Properties

Label 2-399-399.200-c1-0-10
Degree $2$
Conductor $399$
Sign $-0.576 - 0.817i$
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  + (−1.70 + 0.300i)3-s + (−0.347 + 1.96i)4-s + (2.64 − 0.0412i)7-s + (2.81 − 1.02i)9-s − 3.46i·12-s + (−4.23 + 5.05i)13-s + (−3.75 − 1.36i)16-s + (0.5 + 4.33i)19-s + (−4.5 + 0.866i)21-s + (−4.69 + 1.71i)25-s + (−4.49 + 2.59i)27-s + (−0.837 + 5.22i)28-s + (−9.12 + 5.26i)31-s + (1.04 + 5.90i)36-s + (8.60 − 4.96i)37-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (−0.173 + 0.984i)4-s + (0.999 − 0.0155i)7-s + (0.939 − 0.342i)9-s − 0.999i·12-s + (−1.17 + 1.40i)13-s + (−0.939 − 0.342i)16-s + (0.114 + 0.993i)19-s + (−0.981 + 0.188i)21-s + (−0.939 + 0.342i)25-s + (−0.866 + 0.499i)27-s + (−0.158 + 0.987i)28-s + (−1.63 + 0.945i)31-s + (0.173 + 0.984i)36-s + (1.41 − 0.816i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.576 - 0.817i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ -0.576 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367941 + 0.709474i\)
\(L(\frac12)\) \(\approx\) \(0.367941 + 0.709474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.300i)T \)
7 \( 1 + (-2.64 + 0.0412i)T \)
19 \( 1 + (-0.5 - 4.33i)T \)
good2 \( 1 + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.23 - 5.05i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (9.12 - 5.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.60 + 4.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.72 - 1.35i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-11.8 - 9.97i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (5.60 - 6.68i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (2.87 + 16.3i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.37 - 6.52i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-18.7 + 3.30i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.70523585970655539221413402213, −10.96498649770144185751701766976, −9.767970719272416594727092105230, −8.918684836492087152921705950182, −7.62694407910089125066496791469, −7.13743687932381605844285978248, −5.72561555039769464736872031308, −4.64866226987951729075788626075, −3.91765208802344764865081506219, −1.96065256592125456763559193113, 0.59358031708113215263733912654, 2.16291786480285047174298181421, 4.44251918401123595887065048313, 5.24552120008214856527369992234, 5.87774866172489331286439027541, 7.16857651394986494874142776655, 7.986744421432576579073064461788, 9.452625035555504963386886564016, 10.18280320332622148194859005539, 11.04029184261272660277776285195

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