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Properties

Label 2-912-19.12-c2-0-13
Degree $2$
Conductor $912$
Sign $0.703 - 0.710i$
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 + (4.17 + 7.23i)5-s − 4.32·7-s + (1.5 − 2.59i)9-s + 16.6·11-s + (2.07 + 1.20i)13-s + (12.5 + 7.23i)15-s + (2.60 + 4.51i)17-s + (−6.76 − 17.7i)19-s + (−6.48 + 3.74i)21-s + (17.8 − 30.9i)23-s + (−22.3 + 38.7i)25-s − 5.19i·27-s + (35.7 + 20.6i)29-s + 36.8i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.834 + 1.44i)5-s − 0.617·7-s + (0.166 − 0.288i)9-s + 1.50·11-s + (0.159 + 0.0923i)13-s + (0.834 + 0.482i)15-s + (0.153 + 0.265i)17-s + (−0.356 − 0.934i)19-s + (−0.308 + 0.178i)21-s + (0.777 − 1.34i)23-s + (−0.894 + 1.54i)25-s − 0.192i·27-s + (1.23 + 0.712i)29-s + 1.18i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.754722049\)
\(L(\frac12)\) \(\approx\) \(2.754722049\)
\(L(2)\) 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 \( 1 + (6.76 + 17.7i)T \)
good5 \( 1 + (-4.17 - 7.23i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 4.32T + 49T^{2} \)
11 \( 1 - 16.6T + 121T^{2} \)
13 \( 1 + (-2.07 - 1.20i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-2.60 - 4.51i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-17.8 + 30.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-35.7 - 20.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 36.8iT - 961T^{2} \)
37 \( 1 - 0.393iT - 1.36e3T^{2} \)
41 \( 1 + (31.8 - 18.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-28.8 - 49.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (27.9 - 48.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-77.2 - 44.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-45.3 + 26.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.1 + 47.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (85.8 + 49.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (69.5 - 40.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-14.7 - 25.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.3 - 31.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 61.9T + 6.88e3T^{2} \)
89 \( 1 + (-0.168 - 0.0970i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-91.0 + 52.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.00377804610406200634460504675, −9.176530241770253842925081477197, −8.521765485778403532231202644244, −7.02461456370637440689261006042, −6.68251026739458122343967534929, −6.13690652714297456755044723824, −4.54261027801204382323201679646, −3.25301565100012479931877187388, −2.69769494139425886567791346450, −1.35344157380521844232276869654, 0.957781701483161854104454035919, 1.99901358330509595590141519050, 3.53962368022156263848715364973, 4.34291956963371516627656619976, 5.44712492544446244247296043991, 6.16575281863019616895302162149, 7.26783816966223561711067131235, 8.565437062825987811988109247097, 8.893768228057460224916676142504, 9.758644966076010174404305091839

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