Properties

Label 2-912-19.4-c1-0-11
Degree $2$
Conductor $912$
Sign $0.600 - 0.799i$
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  + (0.173 + 0.984i)3-s + (0.907 − 0.761i)5-s + (0.266 − 0.460i)7-s + (−0.939 + 0.342i)9-s + (0.939 + 1.62i)11-s + (−0.673 + 3.82i)13-s + (0.907 + 0.761i)15-s + (−1.09 − 0.397i)17-s + (3.93 − 1.86i)19-s + (0.5 + 0.181i)21-s + (5.13 + 4.30i)23-s + (−0.624 + 3.54i)25-s + (−0.5 − 0.866i)27-s + (3.77 − 1.37i)29-s + (−0.979 + 1.69i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (0.405 − 0.340i)5-s + (0.100 − 0.174i)7-s + (−0.313 + 0.114i)9-s + (0.283 + 0.490i)11-s + (−0.186 + 1.05i)13-s + (0.234 + 0.196i)15-s + (−0.264 − 0.0964i)17-s + (0.903 − 0.427i)19-s + (0.109 + 0.0397i)21-s + (1.07 + 0.898i)23-s + (−0.124 + 0.708i)25-s + (−0.0962 − 0.166i)27-s + (0.701 − 0.255i)29-s + (−0.175 + 0.304i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57937 + 0.789187i\)
\(L(\frac12)\) \(\approx\) \(1.57937 + 0.789187i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.93 + 1.86i)T \)
good5 \( 1 + (-0.907 + 0.761i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.266 + 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.939 - 1.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.673 - 3.82i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.09 + 0.397i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.13 - 4.30i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.77 + 1.37i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.979 - 1.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + (1.56 + 8.84i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.85 - 1.55i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.91 + 0.698i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-9.93 - 8.33i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.51 + 0.916i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.69 + 7.29i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (10.4 - 3.82i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.65 - 3.90i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.0569 + 0.322i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.80 - 15.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.78 + 10.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.618 - 3.50i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.52 + 2.01i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.06498246314466627123080379137, −9.197597557678837991624515590878, −9.023420968542684923118905693039, −7.58005337583023493074127725198, −6.91288697144290212223964113080, −5.71121248313364850409824556373, −4.85763839093879582831536877961, −4.05952900078461909307148297577, −2.80011279300455291717957663008, −1.42477986384020180366577454544, 0.945202638916765697906142328138, 2.46462490348901699069921742642, 3.28619461679563280672285440963, 4.76076498848460182381162928572, 5.80666352341782120216757330676, 6.47238575690644009746254837944, 7.45504500605860259410383067102, 8.259185233121225806382106041913, 9.028747300951869110528177260027, 10.06881685517783487892769755011

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