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Properties

Label 2-912-12.11-c1-0-16
Degree $2$
Conductor $912$
Sign $0.999 + 0.0111i$
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  + (−1.50 + 0.849i)3-s − 0.771i·5-s + 1.48i·7-s + (1.55 − 2.56i)9-s + 4.34·11-s − 5.42·13-s + (0.655 + 1.16i)15-s − 7.36i·17-s + i·19-s + (−1.26 − 2.24i)21-s + 1.65·23-s + 4.40·25-s + (−0.173 + 5.19i)27-s + 5.57i·29-s + 1.97i·31-s + ⋯
L(s)  = 1  + (−0.871 + 0.490i)3-s − 0.345i·5-s + 0.561i·7-s + (0.519 − 0.854i)9-s + 1.31·11-s − 1.50·13-s + (0.169 + 0.300i)15-s − 1.78i·17-s + 0.229i·19-s + (−0.275 − 0.489i)21-s + 0.344·23-s + 0.880·25-s + (−0.0334 + 0.999i)27-s + 1.03i·29-s + 0.354i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0111i$
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: \((2,\ 912,\ (\ :1/2),\ 0.999 + 0.0111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16717 - 0.00650511i\)
\(L(\frac12)\) \(\approx\) \(1.16717 - 0.00650511i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 0.849i)T \)
19 \( 1 - iT \)
good5 \( 1 + 0.771iT - 5T^{2} \)
7 \( 1 - 1.48iT - 7T^{2} \)
11 \( 1 - 4.34T + 11T^{2} \)
13 \( 1 + 5.42T + 13T^{2} \)
17 \( 1 + 7.36iT - 17T^{2} \)
23 \( 1 - 1.65T + 23T^{2} \)
29 \( 1 - 5.57iT - 29T^{2} \)
31 \( 1 - 1.97iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 3.92iT - 41T^{2} \)
43 \( 1 + 8.99iT - 43T^{2} \)
47 \( 1 - 3.62T + 47T^{2} \)
53 \( 1 + 2.72iT - 53T^{2} \)
59 \( 1 - 1.94T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 - 3.32iT - 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 4.94iT - 89T^{2} \)
97 \( 1 - 5.51T + 97T^{2} \)
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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

−9.937149046423017720371742667057, −9.359612655642411024329885981920, −8.760639111474948532091724845318, −7.19600768740497750870936116920, −6.77681271356531374177453286423, −5.44863819961734556310317385905, −4.99203402000722076450208813709, −3.97827668689428976555584054625, −2.60514117227864126594214373197, −0.840529715052881975146053330808, 1.01799169082356786084474876732, 2.37181669005611856633589280411, 3.98180050559774282957072728837, 4.73904290365440930684120110087, 6.01903751125828108031458797645, 6.60269237760964936929926851541, 7.39087807348694346475607566825, 8.192569983750909672593216859696, 9.503634938501165588509045922177, 10.15148795141376070144739282286

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