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Properties

Label 2-912-57.50-c1-0-18
Degree $2$
Conductor $912$
Sign $0.983 + 0.183i$
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  + (1.25 − 1.19i)3-s + (−1.30 + 0.753i)5-s + 3.02·7-s + (0.129 − 2.99i)9-s + 4.73i·11-s + (2.73 + 1.57i)13-s + (−0.729 + 2.50i)15-s + (3.56 − 2.05i)17-s + (−3.27 + 2.87i)19-s + (3.78 − 3.62i)21-s + (5.45 + 3.14i)23-s + (−1.36 + 2.36i)25-s + (−3.42 − 3.90i)27-s + (3.81 − 6.61i)29-s − 7.27i·31-s + ⋯
L(s)  = 1  + (0.722 − 0.691i)3-s + (−0.583 + 0.337i)5-s + 1.14·7-s + (0.0430 − 0.999i)9-s + 1.42i·11-s + (0.757 + 0.437i)13-s + (−0.188 + 0.647i)15-s + (0.864 − 0.499i)17-s + (−0.751 + 0.659i)19-s + (0.826 − 0.791i)21-s + (1.13 + 0.656i)23-s + (−0.272 + 0.472i)25-s + (−0.659 − 0.751i)27-s + (0.708 − 1.22i)29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.983 + 0.183i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14027 - 0.197892i\)
\(L(\frac12)\) \(\approx\) \(2.14027 - 0.197892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 1.19i)T \)
19 \( 1 + (3.27 - 2.87i)T \)
good5 \( 1 + (1.30 - 0.753i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.02T + 7T^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
13 \( 1 + (-2.73 - 1.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.56 + 2.05i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.45 - 3.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.81 + 6.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.27iT - 31T^{2} \)
37 \( 1 + 1.63iT - 37T^{2} \)
41 \( 1 + (5.98 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.45 - 9.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.43 - 1.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.08 - 3.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.79 - 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.68 + 6.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.81 - 1.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.03 + 1.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.24 + 2.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.72 - 3.30i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + (1.41 - 2.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.24 - 3.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.926077177204616969247749135757, −9.139186097149425888273586512381, −8.135330069944894695971282675897, −7.61918622832247700800200627895, −6.99697189745860991725533878485, −5.83045791557716452615945453600, −4.52160775847623957652936186637, −3.72390299422987862263483630989, −2.37254111237939275923871118806, −1.37677398114930400328027580989, 1.20600805292946018003272558393, 2.89508014568260939016617450390, 3.73284369927535545878997292459, 4.75534126682810161911371590038, 5.45081009707887446156600646032, 6.80020202895201487145264310051, 8.150320832318656193535718382010, 8.391891390652840592753369523757, 8.884879199255469669530791019063, 10.33648301581165232309189027955

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