↑

Properties

Label 2-912-76.15-c1-0-8
Degree $2$
Conductor $912$
Sign $0.540 - 0.841i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)3-s + (−0.718 + 0.602i)5-s + (0.983 + 0.567i)7-s + (−0.939 + 0.342i)9-s + (4.11 − 2.37i)11-s + (3.38 + 0.597i)13-s + (−0.718 − 0.602i)15-s + (1.96 + 0.716i)17-s + (−4.19 − 1.18i)19-s + (−0.388 + 1.06i)21-s + (2.10 − 2.51i)23-s + (−0.715 + 4.05i)25-s + (−0.5 − 0.866i)27-s + (1.66 + 4.56i)29-s + (−1.20 + 2.09i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (−0.321 + 0.269i)5-s + (0.371 + 0.214i)7-s + (−0.313 + 0.114i)9-s + (1.24 − 0.717i)11-s + (0.939 + 0.165i)13-s + (−0.185 − 0.155i)15-s + (0.477 + 0.173i)17-s + (−0.962 − 0.272i)19-s + (−0.0847 + 0.232i)21-s + (0.439 − 0.523i)23-s + (−0.143 + 0.811i)25-s + (−0.0962 − 0.166i)27-s + (0.308 + 0.847i)29-s + (−0.217 + 0.376i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.540 - 0.841i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51650 + 0.827901i\)
\(L(\frac12)\) \(\approx\) \(1.51650 + 0.827901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (4.19 + 1.18i)T \)
good5 \( 1 + (0.718 - 0.602i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.983 - 0.567i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.11 + 2.37i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.38 - 0.597i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.96 - 0.716i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.10 + 2.51i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.66 - 4.56i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.20 - 2.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 + (-10.1 + 1.79i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.44 - 1.71i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.471 - 1.29i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (6.30 - 7.51i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.413 + 0.150i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-4.10 - 3.44i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (9.39 - 3.41i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-12.1 + 10.2i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.423 + 2.39i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.81 + 10.2i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.55 + 2.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.703 - 0.123i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.69 + 4.66i)T + (-74.3 - 62.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.38033521483057936736893542562, −9.116215300147311687478261567607, −8.789374216303366003571169448182, −7.86034626687005841871418797099, −6.65536581188903897863378711582, −5.99085353369132885360022662105, −4.79041023784980958259595784346, −3.86318752863848008323019746687, −3.05210561511424103343353598213, −1.36167295566706723649840442600, 0.984929895351577063970481695652, 2.16096300042436489448386984527, 3.74628460796094669151776179940, 4.41641982290799230609407373848, 5.78695195576902254861949334885, 6.55271221025248273208967512909, 7.50344195699495801171210901766, 8.205953948171623892364608795742, 9.034272839870678979392237219333, 9.842167541104022648221079673063

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