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Properties

Label 2-912-19.8-c2-0-14
Degree $2$
Conductor $912$
Sign $-0.346 - 0.937i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (0.328 − 0.568i)5-s + 2.26·7-s + (1.5 + 2.59i)9-s − 1.81·11-s + (−9.14 + 5.27i)13-s + (0.985 − 0.568i)15-s + (−9.38 + 16.2i)17-s + (−13.5 + 13.3i)19-s + (3.39 + 1.96i)21-s + (−0.434 − 0.751i)23-s + (12.2 + 21.2i)25-s + 5.19i·27-s + (−5.65 + 3.26i)29-s − 25.1i·31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (0.0656 − 0.113i)5-s + 0.323·7-s + (0.166 + 0.288i)9-s − 0.164·11-s + (−0.703 + 0.406i)13-s + (0.0656 − 0.0379i)15-s + (−0.552 + 0.956i)17-s + (−0.711 + 0.703i)19-s + (0.161 + 0.0934i)21-s + (−0.0188 − 0.0326i)23-s + (0.491 + 0.851i)25-s + 0.192i·27-s + (−0.194 + 0.112i)29-s − 0.811i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.346 - 0.937i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ -0.346 - 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.622610140\)
\(L(\frac12)\) \(\approx\) \(1.622610140\)
\(L(2)\) 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 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (13.5 - 13.3i)T \)
good5 \( 1 + (-0.328 + 0.568i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 2.26T + 49T^{2} \)
11 \( 1 + 1.81T + 121T^{2} \)
13 \( 1 + (9.14 - 5.27i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (9.38 - 16.2i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (0.434 + 0.751i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (5.65 - 3.26i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 25.1iT - 961T^{2} \)
37 \( 1 - 60.4iT - 1.36e3T^{2} \)
41 \( 1 + (-49.3 - 28.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-20.0 + 34.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-7.05 - 12.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9.28 + 5.36i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (54.3 + 31.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.5 - 63.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (26.6 - 15.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-53.9 - 31.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (59.4 - 102. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (6.11 + 3.53i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 9.32T + 6.88e3T^{2} \)
89 \( 1 + (-25.1 + 14.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (14.3 + 8.28i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.08513171073072921911936811550, −9.307375562001907292049978932495, −8.473565736662253661147611789120, −7.80228222692491584997102699108, −6.79636606917441204251247247595, −5.78073246518707758740459968590, −4.68739645884384687305014013074, −3.94483330939137157120444536694, −2.65913914157185343286799579632, −1.58656913229719470205283097502, 0.47017420762972144699307859603, 2.14272560969719925862212673733, 2.91545894334934613929010520015, 4.29728543377182724387765981481, 5.13240173543128027016103101223, 6.33218798399605366608401180726, 7.21440457357881020962726340128, 7.87653270967348086263155084100, 8.879132855941849330777325923890, 9.444977422309738294100001979190

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