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Properties

Label 2-912-57.47-c0-0-0
Degree $2$
Conductor $912$
Sign $0.776 + 0.630i$
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  + (−0.766 + 0.642i)3-s + (0.766 − 1.32i)7-s + (0.173 − 0.984i)9-s + (−1.43 − 1.20i)13-s + (0.5 − 0.866i)19-s + (0.266 + 1.50i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (0.173 − 0.300i)31-s + 0.347·37-s + 1.87·39-s + (0.326 + 0.118i)43-s + (−0.673 − 1.16i)49-s + (0.173 + 0.984i)57-s + (1.76 − 0.642i)61-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.766 − 1.32i)7-s + (0.173 − 0.984i)9-s + (−1.43 − 1.20i)13-s + (0.5 − 0.866i)19-s + (0.266 + 1.50i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (0.173 − 0.300i)31-s + 0.347·37-s + 1.87·39-s + (0.326 + 0.118i)43-s + (−0.673 − 1.16i)49-s + (0.173 + 0.984i)57-s + (1.76 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7785090024\)
\(L(\frac12)\) \(\approx\) \(0.7785090024\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.347T + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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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

−10.14785017164775717385138509866, −9.784322581990235479980890873378, −8.542212332971842085845523322022, −7.41644885854026472769038403821, −7.01400931352822952037165095139, −5.58203363676118635319975095362, −4.90754189264064532091556541060, −4.15852599832802457999956165105, −2.91483501260857518765720066120, −0.897879797936330743191016736577, 1.73102939741099338568308066525, 2.59443019736734478909813973360, 4.51270146605892299673393448812, 5.19202263432930426657095381939, 6.02897258846825674796587647910, 6.96005235402756335937360592385, 7.78288830874562989212292478032, 8.643723776672381609071210553169, 9.538234560475151885827212757006, 10.48889852050092989360402213402

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