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Properties

Label 2-912-228.59-c0-0-1
Degree $2$
Conductor $912$
Sign $0.624 + 0.780i$
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 + (1.11 − 0.642i)7-s + (0.173 + 0.984i)9-s + (0.439 + 0.524i)13-s + (−0.5 − 0.866i)19-s + (−1.26 − 0.223i)21-s + (0.766 − 0.642i)25-s + (0.500 − 0.866i)27-s + (−0.173 − 0.300i)31-s − 1.96i·37-s − 0.684i·39-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (−1.76 − 0.642i)61-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (1.11 − 0.642i)7-s + (0.173 + 0.984i)9-s + (0.439 + 0.524i)13-s + (−0.5 − 0.866i)19-s + (−1.26 − 0.223i)21-s + (0.766 − 0.642i)25-s + (0.500 − 0.866i)27-s + (−0.173 − 0.300i)31-s − 1.96i·37-s − 0.684i·39-s + (0.673 + 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (−1.76 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.624 + 0.780i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8865134173\)
\(L(\frac12)\) \(\approx\) \(0.8865134173\)
\(L(1)\) 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 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
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 \( 1 + 1.96iT - T^{2} \)
41 \( 1 + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
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 \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \)
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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

−10.58372721198729268980463675052, −9.290937859256195661131792514391, −8.314366657822335972157386743959, −7.55588288845736391826192162330, −6.80407470476834646452168645921, −5.92167410219627062952952555269, −4.84743192540101352413871931320, −4.19085565706533795696502256892, −2.37304353799452129710216815034, −1.15678897074856065061015346706, 1.55189613574610043673277534186, 3.19907368848410111279421366728, 4.36106983227531909691483632594, 5.21575644356347438825486665760, 5.85187099726469081950552789103, 6.87450484162422450138840103414, 8.088580974257202841202410105078, 8.726701956571249256715200532494, 9.664023848962462170502391955582, 10.67921681640189324714043900808

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