↑

Properties

Label 2-912-12.11-c1-0-26
Degree $2$
Conductor $912$
Sign $0.896 + 0.443i$
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  + (1.72 − 0.110i)3-s − 3.67i·5-s + 4.87i·7-s + (2.97 − 0.383i)9-s − 0.458·11-s + 4.96·13-s + (−0.407 − 6.35i)15-s − 3.91i·17-s − i·19-s + (0.540 + 8.42i)21-s + 2.77·23-s − 8.50·25-s + (5.10 − 0.992i)27-s + 4.21i·29-s + 3.33i·31-s + ⋯
L(s)  = 1  + (0.997 − 0.0639i)3-s − 1.64i·5-s + 1.84i·7-s + (0.991 − 0.127i)9-s − 0.138·11-s + 1.37·13-s + (−0.105 − 1.64i)15-s − 0.949i·17-s − 0.229i·19-s + (0.117 + 1.83i)21-s + 0.577·23-s − 1.70·25-s + (0.981 − 0.190i)27-s + 0.782i·29-s + 0.599i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32530 - 0.543915i\)
\(L(\frac12)\) \(\approx\) \(2.32530 - 0.543915i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.110i)T \)
19 \( 1 + iT \)
good5 \( 1 + 3.67iT - 5T^{2} \)
7 \( 1 - 4.87iT - 7T^{2} \)
11 \( 1 + 0.458T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 + 3.91iT - 17T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 - 4.21iT - 29T^{2} \)
31 \( 1 - 3.33iT - 31T^{2} \)
37 \( 1 - 6.65T + 37T^{2} \)
41 \( 1 + 9.60iT - 41T^{2} \)
43 \( 1 + 6.41iT - 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 3.95iT - 53T^{2} \)
59 \( 1 + 2.01T + 59T^{2} \)
61 \( 1 + 6.63T + 61T^{2} \)
67 \( 1 - 1.01iT - 67T^{2} \)
71 \( 1 - 5.71T + 71T^{2} \)
73 \( 1 - 9.95T + 73T^{2} \)
79 \( 1 + 6.29iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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

−9.358879887904957022121103479384, −9.167611304094235200078086286512, −8.546803967711844101523932489450, −7.955715588516389206551542953845, −6.56537002986732231853803306114, −5.42529255550544791330543104609, −4.86560328966651698259569194665, −3.56110856934628746809621988326, −2.42132771369122834242247216503, −1.28254457979577077443686971562, 1.46033651669302945333310574172, 2.95935004815461864991828222782, 3.67003665442279874973261133830, 4.33621582884199057233825667498, 6.33710258717204801916844072216, 6.71632748579524238213090280975, 7.86135720576469310003416988737, 8.025995284711238336116706360047, 9.613827809623824386999897487881, 10.12756213711631675336708814778

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