Properties

Label 2-912-19.9-c1-0-3
Degree $2$
Conductor $912$
Sign $-0.660 - 0.750i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−0.766 + 0.642i)3-s + (0.826 + 0.300i)5-s + (−1.09 + 1.89i)7-s + (0.173 − 0.984i)9-s + (−0.0812 − 0.140i)11-s + (0.581 + 0.487i)13-s + (−0.826 + 0.300i)15-s + (0.539 + 3.05i)17-s + (−2.77 + 3.35i)19-s + (−0.379 − 2.15i)21-s + (1.21 − 0.441i)23-s + (−3.23 − 2.71i)25-s + (0.500 + 0.866i)27-s + (−1.13 + 6.41i)29-s + (0.479 − 0.829i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (0.369 + 0.134i)5-s + (−0.412 + 0.715i)7-s + (0.0578 − 0.328i)9-s + (−0.0244 − 0.0424i)11-s + (0.161 + 0.135i)13-s + (−0.213 + 0.0776i)15-s + (0.130 + 0.741i)17-s + (−0.637 + 0.770i)19-s + (−0.0827 − 0.469i)21-s + (0.252 − 0.0920i)23-s + (−0.647 − 0.543i)25-s + (0.0962 + 0.166i)27-s + (−0.210 + 1.19i)29-s + (0.0860 − 0.149i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.660 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380806 + 0.841970i\)
\(L(\frac12)\) \(\approx\) \(0.380806 + 0.841970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (2.77 - 3.35i)T \)
good5 \( 1 + (-0.826 - 0.300i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.09 - 1.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0812 + 0.140i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.581 - 0.487i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.539 - 3.05i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.21 + 0.441i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.13 - 6.41i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.479 + 0.829i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.16T + 37T^{2} \)
41 \( 1 + (8.11 - 6.81i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.166 + 0.0605i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.602 + 3.41i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.83 - 2.84i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.482 - 2.73i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.79 - 2.47i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.184 - 1.04i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.77 - 1.73i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.72 + 1.44i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.01 + 3.36i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.55 - 14.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.9 - 10.0i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.63 + 9.27i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.37249278918257437919959794781, −9.661225502356515891832788680432, −8.820668859858642979862119661850, −8.018627475202872402148117456348, −6.68899501514532183507347488276, −6.07353418931826029497743160913, −5.29755353569939446277988069589, −4.14915968120378601622293514389, −3.08223815905244983724436294723, −1.73579350098789440633448833991, 0.45127061799494670711253750165, 1.96275492818626141448336595777, 3.33291841155477083199561106358, 4.52456741605647699539850553082, 5.46715402335344388622429710712, 6.42229899928949838657070188334, 7.12606093827210158249149632005, 7.962800367920204892612832014311, 9.072535565189332411995962647181, 9.835691954225003538141970577895

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