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Properties

Label 2-912-19.6-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.994 - 0.108i$
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  + (0.939 + 0.342i)3-s + (−0.0812 − 0.460i)5-s + (−2.20 + 3.82i)7-s + (0.766 + 0.642i)9-s + (−2.76 − 4.79i)11-s + (−5.62 + 2.04i)13-s + (0.0812 − 0.460i)15-s + (−3.33 + 2.79i)17-s + (−4.34 − 0.405i)19-s + (−3.37 + 2.83i)21-s + (0.549 − 3.11i)23-s + (4.49 − 1.63i)25-s + (0.500 + 0.866i)27-s + (−1.15 − 0.970i)29-s + (−1.09 + 1.89i)31-s + ⋯
L(s)  = 1  + (0.542 + 0.197i)3-s + (−0.0363 − 0.206i)5-s + (−0.833 + 1.44i)7-s + (0.255 + 0.214i)9-s + (−0.833 − 1.44i)11-s + (−1.55 + 0.567i)13-s + (0.0209 − 0.118i)15-s + (−0.807 + 0.677i)17-s + (−0.995 − 0.0929i)19-s + (−0.737 + 0.618i)21-s + (0.114 − 0.649i)23-s + (0.898 − 0.327i)25-s + (0.0962 + 0.166i)27-s + (−0.214 − 0.180i)29-s + (−0.196 + 0.339i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.994 - 0.108i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0196895 + 0.363417i\)
\(L(\frac12)\) \(\approx\) \(0.0196895 + 0.363417i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (4.34 + 0.405i)T \)
good5 \( 1 + (0.0812 + 0.460i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.20 - 3.82i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.76 + 4.79i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.62 - 2.04i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.33 - 2.79i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.549 + 3.11i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.15 + 0.970i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.09 - 1.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.75T + 37T^{2} \)
41 \( 1 + (-1.84 - 0.669i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.624 + 3.54i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.08 - 5.94i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.464 - 2.63i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.01 - 0.853i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.15 - 6.53i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.90 - 1.60i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.31 - 7.48i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.97 + 1.08i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.18 - 0.432i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (8.96 - 15.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.7 - 4.26i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (6.36 - 5.34i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.42993364971206464838933501582, −9.477205540912730737052840482523, −8.689662814920686207354478557708, −8.428596060766844519345542077125, −7.06027983275433055071221577814, −6.14097036971067358739421288807, −5.29883166420315531188274808681, −4.23925861648416758687646683432, −2.83255445815569259769939776845, −2.38314408311264575334568649225, 0.14364942974140608245842446982, 2.13685416197469883838336084077, 3.10209072734329446410424156500, 4.28776364103278166554268841014, 5.04151066397543972740025218449, 6.63730268258825506286247912239, 7.35823815549558153773647174470, 7.55108557383870780663741284534, 8.978526278490056112511272107040, 9.934777126491318166873165629667

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