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Properties

Label 2-912-19.12-c2-0-17
Degree $2$
Conductor $912$
Sign $0.666 - 0.745i$
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 + (2.76 + 4.79i)5-s + 7.80·7-s + (1.5 − 2.59i)9-s + 8.70·11-s + (16.0 + 9.28i)13-s + (−8.29 − 4.79i)15-s + (7.30 + 12.6i)17-s + (−7.66 − 17.3i)19-s + (−11.7 + 6.76i)21-s + (16.4 − 28.5i)23-s + (−2.80 + 4.85i)25-s + 5.19i·27-s + (0.900 + 0.519i)29-s − 19.5i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.553 + 0.958i)5-s + 1.11·7-s + (0.166 − 0.288i)9-s + 0.791·11-s + (1.23 + 0.714i)13-s + (−0.553 − 0.319i)15-s + (0.429 + 0.744i)17-s + (−0.403 − 0.914i)19-s + (−0.557 + 0.322i)21-s + (0.716 − 1.24i)23-s + (−0.112 + 0.194i)25-s + 0.192i·27-s + (0.0310 + 0.0179i)29-s − 0.631i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.666 - 0.745i$
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: \((2,\ 912,\ (\ :1),\ 0.666 - 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.357955935\)
\(L(\frac12)\) \(\approx\) \(2.357955935\)
\(L(2)\) 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 \( 1 + (7.66 + 17.3i)T \)
good5 \( 1 + (-2.76 - 4.79i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 7.80T + 49T^{2} \)
11 \( 1 - 8.70T + 121T^{2} \)
13 \( 1 + (-16.0 - 9.28i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-7.30 - 12.6i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-16.4 + 28.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.900 - 0.519i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 19.5iT - 961T^{2} \)
37 \( 1 + 30.2iT - 1.36e3T^{2} \)
41 \( 1 + (19.7 - 11.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-0.142 - 0.246i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-30.1 + 52.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-85.4 - 49.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (66.9 - 38.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (19.6 - 34.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-53.0 - 30.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (93.0 - 53.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (45.7 + 79.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (1.09 - 0.631i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 27.6T + 6.88e3T^{2} \)
89 \( 1 + (-57.3 - 33.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (76.4 - 44.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.35250575199308073315004159804, −9.062215434513772118992975309732, −8.559666288455506565905069622960, −7.24659449816092192322318118035, −6.46716294316009371223295077092, −5.84368114459968825067942255393, −4.62258768138127453296086447531, −3.83026285817628431115788804509, −2.39107433151794920595490002821, −1.20330003993698814361964444303, 1.10509165929146098704882031912, 1.59569072644337684855429383939, 3.45813980099322944125213199950, 4.70617702915444362826744355753, 5.41444467340625860420067193754, 6.10627512115773034893906522896, 7.28771556386719930172960243584, 8.215410182140085666803539366690, 8.836213883571401320578237444463, 9.744034567643404588009279063795

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