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Properties

Label 2-912-19.8-c2-0-16
Degree $2$
Conductor $912$
Sign $0.730 + 0.683i$
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 + (−3.13 + 5.42i)5-s − 8.36·7-s + (1.5 + 2.59i)9-s − 16.7·11-s + (−14.4 + 8.36i)13-s + (9.39 − 5.42i)15-s + (−0.0962 + 0.166i)17-s + (6.09 − 17.9i)19-s + (12.5 + 7.24i)21-s + (2.77 + 4.79i)23-s + (−7.12 − 12.3i)25-s − 5.19i·27-s + (13.4 − 7.76i)29-s + 30.6i·31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.626 + 1.08i)5-s − 1.19·7-s + (0.166 + 0.288i)9-s − 1.52·11-s + (−1.11 + 0.643i)13-s + (0.626 − 0.361i)15-s + (−0.00566 + 0.00981i)17-s + (0.320 − 0.947i)19-s + (0.597 + 0.344i)21-s + (0.120 + 0.208i)23-s + (−0.285 − 0.493i)25-s − 0.192i·27-s + (0.464 − 0.267i)29-s + 0.987i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.730 + 0.683i$
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: \((2,\ 912,\ (\ :1),\ 0.730 + 0.683i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3991065171\)
\(L(\frac12)\) \(\approx\) \(0.3991065171\)
\(L(2)\) 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 \( 1 + (-6.09 + 17.9i)T \)
good5 \( 1 + (3.13 - 5.42i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 8.36T + 49T^{2} \)
11 \( 1 + 16.7T + 121T^{2} \)
13 \( 1 + (14.4 - 8.36i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (0.0962 - 0.166i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-2.77 - 4.79i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.4 + 7.76i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 30.6iT - 961T^{2} \)
37 \( 1 + 65.6iT - 1.36e3T^{2} \)
41 \( 1 + (-15.8 - 9.16i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4.63 - 8.02i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-44.4 - 76.9i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (2.56 - 1.48i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (52.3 + 30.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.1 - 59.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (36.4 - 21.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-49.9 - 28.8i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-25.0 + 43.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (21.5 + 12.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 61.2T + 6.88e3T^{2} \)
89 \( 1 + (-98.6 + 56.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (98.4 + 56.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.952512841185544224067170551979, −9.117894296023372692902669042174, −7.68888873976979597277123980686, −7.23940051641108665421515231320, −6.54990208731175361234263708747, −5.52005532961709348054474011163, −4.47416785607505729064187285461, −3.10790129483976897047339023231, −2.50950481988537034761954936564, −0.24338443559636853300526967186, 0.61066166853026520251930941722, 2.63322131042590885084907105805, 3.72117109229458029893500155602, 4.89456920834726564924874899126, 5.38712952560842743186952594124, 6.48088721818284938522086881334, 7.63862149102289870381175048010, 8.196647631672617726870743793597, 9.313946344714500070755004424452, 10.07945730937633423222773048764

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