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Properties

Label 2-912-76.11-c2-0-12
Degree $2$
Conductor $912$
Sign $0.516 - 0.856i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−4.96 + 8.60i)5-s − 5.53i·7-s + (1.5 + 2.59i)9-s − 15.2i·11-s + (8.32 + 14.4i)13-s + (14.8 − 8.60i)15-s + (9.95 − 17.2i)17-s + (4.37 − 18.4i)19-s + (−4.79 + 8.30i)21-s + (−19.9 + 11.5i)23-s + (−36.8 − 63.7i)25-s − 5.19i·27-s + (10.7 + 18.5i)29-s + 24.1i·31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.993 + 1.72i)5-s − 0.791i·7-s + (0.166 + 0.288i)9-s − 1.38i·11-s + (0.640 + 1.10i)13-s + (0.993 − 0.573i)15-s + (0.585 − 1.01i)17-s + (0.230 − 0.973i)19-s + (−0.228 + 0.395i)21-s + (−0.867 + 0.500i)23-s + (−1.47 − 2.55i)25-s − 0.192i·27-s + (0.369 + 0.639i)29-s + 0.779i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.516 - 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.075459150\)
\(L(\frac12)\) \(\approx\) \(1.075459150\)
\(L(2)\) 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 \( 1 + (1.5 + 0.866i)T \)
19 \( 1 + (-4.37 + 18.4i)T \)
good5 \( 1 + (4.96 - 8.60i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 5.53iT - 49T^{2} \)
11 \( 1 + 15.2iT - 121T^{2} \)
13 \( 1 + (-8.32 - 14.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-9.95 + 17.2i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (19.9 - 11.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.7 - 18.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 24.1iT - 961T^{2} \)
37 \( 1 - 27.5T + 1.36e3T^{2} \)
41 \( 1 + (32.6 - 56.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-68.3 - 39.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (34.7 - 20.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.3 - 66.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.1 - 16.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (38.5 + 66.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (31.2 - 18.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (69.7 + 40.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-18.8 + 32.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-54.8 - 31.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 25.1iT - 6.88e3T^{2} \)
89 \( 1 + (-50.9 - 88.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-50.1 + 86.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.40099532488951250899312916972, −9.254891057923768929943264049204, −8.012284204739201056224811906740, −7.39410522912501486235646369721, −6.65617275895474018277490303883, −6.06605384741860476860659237443, −4.54739400650251680003948340343, −3.55692537564761737426385589830, −2.82177080095430905505730307982, −0.892114974449010542419877006204, 0.52431531953462650812999385333, 1.83866114824186813310954928195, 3.77616827619045582922596830712, 4.35544324212286272637755509249, 5.44511707307990175208099389530, 5.86525278306813314115065482451, 7.50343065012346205406704925678, 8.179947007841187937147029920872, 8.783242917128752277564333225407, 9.794859927787739811199804407962

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