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Properties

Label 2-912-228.167-c0-0-1
Degree $2$
Conductor $912$
Sign $-0.973 + 0.226i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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 + (−1.70 − 0.984i)7-s + (−0.939 + 0.342i)9-s + (−1.26 − 0.223i)13-s + (−0.5 + 0.866i)19-s + (−0.673 + 1.85i)21-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)27-s + (0.939 − 1.62i)31-s − 0.684i·37-s + 1.28i·39-s + (−0.439 − 0.524i)43-s + (1.43 + 2.49i)49-s + (0.939 + 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−1.70 − 0.984i)7-s + (−0.939 + 0.342i)9-s + (−1.26 − 0.223i)13-s + (−0.5 + 0.866i)19-s + (−0.673 + 1.85i)21-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)27-s + (0.939 − 1.62i)31-s − 0.684i·37-s + 1.28i·39-s + (−0.439 − 0.524i)43-s + (1.43 + 2.49i)49-s + (0.939 + 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.973 + 0.226i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ -0.973 + 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4433733336\)
\(L(\frac12)\) \(\approx\) \(0.4433733336\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.684iT - T^{2} \)
41 \( 1 + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
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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

−10.04448951972359611999194701014, −9.154115592471596509797351968645, −7.919518738328860784594128959819, −7.34354506136687847815291887619, −6.46716104839941629485766120737, −5.95359183853566684772031898261, −4.51051428383263866471980430663, −3.32990757068716964659162970282, −2.29719769597779039060303529813, −0.40941572019822268493642859116, 2.66275485607305967635879276549, 3.24878020292478421024526682312, 4.58166578572640032846694719431, 5.39127488291137523278051022394, 6.34942413328443025463491112187, 7.04703246177559855001635527610, 8.574355215691307099945988370922, 9.179325682124157940831977577333, 9.811789848593066648696391352612, 10.40157355418931273539687752875

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