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Properties

Label 2-912-228.23-c1-0-7
Degree $2$
Conductor $912$
Sign $0.932 - 0.361i$
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  + (−1.70 + 0.300i)3-s + (−2.89 − 1.66i)7-s + (2.81 − 1.02i)9-s + (−1.03 + 5.89i)13-s + (0.5 − 4.33i)19-s + (5.43 + 1.97i)21-s + (0.868 − 4.92i)25-s + (−4.49 + 2.59i)27-s + (9.64 + 5.56i)31-s + 11.7·37-s − 10.3i·39-s + (4.51 + 5.38i)43-s + (2.07 + 3.58i)49-s + (0.449 + 7.53i)57-s + (11.6 + 9.75i)61-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (−1.09 − 0.630i)7-s + (0.939 − 0.342i)9-s + (−0.288 + 1.63i)13-s + (0.114 − 0.993i)19-s + (1.18 + 0.431i)21-s + (0.173 − 0.984i)25-s + (−0.866 + 0.499i)27-s + (1.73 + 0.999i)31-s + 1.93·37-s − 1.66i·39-s + (0.689 + 0.821i)43-s + (0.295 + 0.512i)49-s + (0.0595 + 0.998i)57-s + (1.48 + 1.24i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.932 - 0.361i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887972 + 0.166021i\)
\(L(\frac12)\) \(\approx\) \(0.887972 + 0.166021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.300i)T \)
19 \( 1 + (-0.5 + 4.33i)T \)
good5 \( 1 + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.89 + 1.66i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.03 - 5.89i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-9.64 - 5.56i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.51 - 5.38i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-11.6 - 9.75i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-5.18 - 14.2i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.216 + 1.22i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.917 - 0.161i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (17.8 + 6.49i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.953977383757901938729281196340, −9.700422908222456070441674660870, −8.605406135339073689662292743061, −7.17223387502420662004646440861, −6.71068130145136740234596622188, −6.01343817530450861782909158136, −4.63939527481145064894417502345, −4.15733215849767412402357401885, −2.69239783612525146253939886885, −0.875539479802654436237841492202, 0.71136562567476585456806275471, 2.52687101826999260512043405356, 3.65874551942728131070283405929, 4.99359687724531524334898458114, 5.85770615773229139320726244979, 6.30964097018816915395654581820, 7.48258213637923011594674548313, 8.189319670544865640801259509452, 9.589066252647156463669434897558, 9.956119625693879550177620730330

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