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Properties

Label 2-912-57.8-c1-0-16
Degree $2$
Conductor $912$
Sign $0.440 - 0.897i$
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  + (−0.772 + 1.55i)3-s + (1.67 + 0.966i)5-s + 1.30·7-s + (−1.80 − 2.39i)9-s + 0.793i·11-s + (5.90 − 3.41i)13-s + (−2.79 + 1.84i)15-s + (3.61 + 2.08i)17-s + (−0.402 + 4.34i)19-s + (−1.00 + 2.01i)21-s + (3.32 − 1.92i)23-s + (−0.631 − 1.09i)25-s + (5.10 − 0.954i)27-s + (0.449 + 0.778i)29-s − 5.18i·31-s + ⋯
L(s)  = 1  + (−0.445 + 0.895i)3-s + (0.748 + 0.432i)5-s + 0.491·7-s + (−0.602 − 0.798i)9-s + 0.239i·11-s + (1.63 − 0.946i)13-s + (−0.720 + 0.477i)15-s + (0.875 + 0.505i)17-s + (−0.0923 + 0.995i)19-s + (−0.219 + 0.440i)21-s + (0.693 − 0.400i)23-s + (−0.126 − 0.218i)25-s + (0.982 − 0.183i)27-s + (0.0834 + 0.144i)29-s − 0.930i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47796 + 0.920604i\)
\(L(\frac12)\) \(\approx\) \(1.47796 + 0.920604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.772 - 1.55i)T \)
19 \( 1 + (0.402 - 4.34i)T \)
good5 \( 1 + (-1.67 - 0.966i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 - 0.793iT - 11T^{2} \)
13 \( 1 + (-5.90 + 3.41i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.61 - 2.08i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.32 + 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.449 - 0.778i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.18iT - 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 + (4.52 - 7.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.92 - 6.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.48 - 3.74i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.22 - 7.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.84 + 8.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 2.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.5 + 6.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.99 - 8.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.83 - 6.64i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 0.580i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + (1.30 + 2.26i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.72 + 1.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16239792707974261412095646384, −9.770132489362582087083940242232, −8.538905705731227814052087803363, −7.966687411736835158674265816459, −6.38978803818170006712832799211, −5.94741609548404613864204778993, −5.07412990492509099137210265892, −3.91386568353167577804136060231, −3.02467713531098728787869584496, −1.35480308527749603010285271319, 1.08942788230565345665537682276, 1.94922793526343452118615498586, 3.45523627821293316972226642104, 4.99819179269353971552032685257, 5.55302626802900971653132708212, 6.56560653868101398254151216194, 7.19554211057149702024002602371, 8.467447292383868138616326378504, 8.825582407590211473877902360446, 9.958590791540677861377042816295

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