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Properties

Label 2-912-228.119-c1-0-21
Degree $2$
Conductor $912$
Sign $0.825 + 0.564i$
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 + (3.93 − 2.27i)7-s + (2.81 + 1.02i)9-s + (0.507 + 2.87i)13-s + (−3.5 − 2.59i)19-s + (−7.39 + 2.68i)21-s + (0.868 + 4.92i)25-s + (−4.49 − 2.59i)27-s + (7.27 − 4.19i)31-s + 7.02·37-s − 5.05i·39-s + (8.03 − 9.57i)43-s + (6.80 − 11.7i)49-s + (5.18 + 5.48i)57-s + (4.80 − 4.03i)61-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (1.48 − 0.858i)7-s + (0.939 + 0.342i)9-s + (0.140 + 0.797i)13-s + (−0.802 − 0.596i)19-s + (−1.61 + 0.586i)21-s + (0.173 + 0.984i)25-s + (−0.866 − 0.499i)27-s + (1.30 − 0.754i)31-s + 1.15·37-s − 0.810i·39-s + (1.22 − 1.46i)43-s + (0.972 − 1.68i)49-s + (0.687 + 0.726i)57-s + (0.615 − 0.516i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29889 - 0.401500i\)
\(L(\frac12)\) \(\approx\) \(1.29889 - 0.401500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 2.59i)T \)
good5 \( 1 + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.93 + 2.27i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.507 - 2.87i)T + (-12.2 + 4.44i)T^{2} \)
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 \( 1 + (-7.27 + 4.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.02T + 37T^{2} \)
41 \( 1 + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-8.03 + 9.57i)T + (-7.46 - 42.3i)T^{2} \)
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 \( 1 + (-4.80 + 4.03i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (5.59 - 15.3i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.87 + 16.3i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-6.84 - 1.20i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.69 + 1.71i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.30489481851537767173690040757, −9.220911338158503418315556377474, −8.152305995493758552603675444258, −7.38231738136167684504504246642, −6.65326542195124966694702665098, −5.60058199761723356052387185821, −4.61173064175108512620433479638, −4.13265430857329472101969863962, −2.09152640630741846394514312315, −0.939596367726316550875704512532, 1.17448830980954796030095028679, 2.53418342428528841579776235084, 4.23318282450304882464179375481, 4.93034679147730976527107135942, 5.78187632521854719738717289625, 6.48307110637357665165552092498, 7.85530812790567550546680609881, 8.304696736734973132052714556266, 9.416337793560701023176653263493, 10.45300998627249316662982691059

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