↑

Properties

Label 2-912-57.2-c1-0-2
Degree $2$
Conductor $912$
Sign $0.490 - 0.871i$
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  + (0.0553 − 1.73i)3-s + (0.882 + 2.42i)5-s + (1.58 + 2.74i)7-s + (−2.99 − 0.191i)9-s + (−2.16 − 1.25i)11-s + (2.71 + 3.24i)13-s + (4.24 − 1.39i)15-s + (−1.32 − 0.233i)17-s + (−3.14 + 3.01i)19-s + (4.83 − 2.59i)21-s + (−1.30 + 3.58i)23-s + (−1.27 + 1.06i)25-s + (−0.497 + 5.17i)27-s + (1.32 + 7.49i)29-s + (−6.89 + 3.97i)31-s + ⋯
L(s)  = 1  + (0.0319 − 0.999i)3-s + (0.394 + 1.08i)5-s + (0.598 + 1.03i)7-s + (−0.997 − 0.0638i)9-s + (−0.653 − 0.377i)11-s + (0.754 + 0.898i)13-s + (1.09 − 0.359i)15-s + (−0.320 − 0.0565i)17-s + (−0.722 + 0.691i)19-s + (1.05 − 0.565i)21-s + (−0.272 + 0.747i)23-s + (−0.254 + 0.213i)25-s + (−0.0956 + 0.995i)27-s + (0.245 + 1.39i)29-s + (−1.23 + 0.714i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28653 + 0.751766i\)
\(L(\frac12)\) \(\approx\) \(1.28653 + 0.751766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0553 + 1.73i)T \)
19 \( 1 + (3.14 - 3.01i)T \)
good5 \( 1 + (-0.882 - 2.42i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.58 - 2.74i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.16 + 1.25i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.71 - 3.24i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.32 + 0.233i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (1.30 - 3.58i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.32 - 7.49i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.89 - 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.10iT - 37T^{2} \)
41 \( 1 + (4.95 + 4.16i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-11.7 + 4.27i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-6.16 + 1.08i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.46 - 1.26i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.54 - 8.75i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.133 + 0.0485i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-4.48 + 0.791i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.59 + 3.12i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (1.67 + 1.40i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (6.41 - 7.64i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-12.3 + 7.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-12.7 + 10.6i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.538 - 0.0949i)T + (91.1 + 33.1i)T^{2} \)
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

−10.64101916826115410472608630004, −9.007383096697430503725716999606, −8.682682928757124399134002596794, −7.56532305078651468430967881648, −6.83877715874667440904749749758, −5.95684107848704729305264540120, −5.38484696566000372260602930546, −3.63851543687247838380164703675, −2.47388760661996238921658835490, −1.77022027099638373468198418704, 0.69768542400036483438792146881, 2.39584681255513760501852428876, 3.92226806087310723218668187717, 4.58723760998333344169086323569, 5.31375722726694380530095511311, 6.28553880545109697692407793835, 7.75712106857823207884946424810, 8.332904710454367453527840372507, 9.172089089764911077523953233673, 9.982506214657916829258988635397

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