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Properties

Label 2-912-76.31-c1-0-16
Degree $2$
Conductor $912$
Sign $-0.397 + 0.917i$
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  + (0.5 + 0.866i)3-s + (−0.912 − 1.58i)5-s − 4.99i·7-s + (−0.499 + 0.866i)9-s + 3.82i·11-s + (1.00 + 0.581i)13-s + (0.912 − 1.58i)15-s + (−3.73 − 6.47i)17-s + (−3.22 + 2.92i)19-s + (4.32 − 2.49i)21-s + (−2.24 − 1.29i)23-s + (0.833 − 1.44i)25-s − 0.999·27-s + (−6.63 − 3.82i)29-s − 0.158·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.408 − 0.707i)5-s − 1.88i·7-s + (−0.166 + 0.288i)9-s + 1.15i·11-s + (0.279 + 0.161i)13-s + (0.235 − 0.408i)15-s + (−0.906 − 1.56i)17-s + (−0.740 + 0.671i)19-s + (0.943 − 0.544i)21-s + (−0.468 − 0.270i)23-s + (0.166 − 0.288i)25-s − 0.192·27-s + (−1.23 − 0.711i)29-s − 0.0285·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577254 - 0.879063i\)
\(L(\frac12)\) \(\approx\) \(0.577254 - 0.879063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (3.22 - 2.92i)T \)
good5 \( 1 + (0.912 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.99iT - 7T^{2} \)
11 \( 1 - 3.82iT - 11T^{2} \)
13 \( 1 + (-1.00 - 0.581i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.73 + 6.47i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.24 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.63 + 3.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.158T + 31T^{2} \)
37 \( 1 + 8.25iT - 37T^{2} \)
41 \( 1 + (1.07 - 0.617i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.90 + 1.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0858 + 0.0495i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.82 - 1.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.14 - 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.64 + 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.97 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.91 - 3.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.32 + 4.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.74 - 8.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.83iT - 83T^{2} \)
89 \( 1 + (-11.9 - 6.87i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.50 + 4.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.793284268725398325124762871744, −9.151484513141202472546891215885, −8.047407573430421627293167706025, −7.39705369930008489966087505519, −6.62528688438457530750834242999, −5.06507161482084384361982587499, −4.23542161926562645513004164028, −3.89578626331130656264703065244, −2.12641758816385493846149897216, −0.46028971149832108218267522082, 1.91479199286248973755654268948, 2.87039429490317797491541952840, 3.77592921552178674439235497726, 5.38310228110193962329685711334, 6.13896174452273004617519172569, 6.78482181588028160027439137724, 8.120666241152668450677159952949, 8.583954561063908068774028296347, 9.184079546775394794748317381631, 10.54130066426787426662897701738

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