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Properties

Label 2-912-19.12-c2-0-10
Degree $2$
Conductor $912$
Sign $0.940 - 0.338i$
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 + (−0.140 − 0.243i)5-s + 5.24·7-s + (1.5 − 2.59i)9-s + 1.15·11-s + (−6.32 − 3.65i)13-s + (0.421 + 0.243i)15-s + (7.52 + 13.0i)17-s + (−1.52 + 18.9i)19-s + (−7.86 + 4.53i)21-s + (13.3 − 23.1i)23-s + (12.4 − 21.5i)25-s + 5.19i·27-s + (−7.76 − 4.48i)29-s + 11.7i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.0280 − 0.0486i)5-s + 0.748·7-s + (0.166 − 0.288i)9-s + 0.105·11-s + (−0.486 − 0.280i)13-s + (0.0280 + 0.0162i)15-s + (0.442 + 0.766i)17-s + (−0.0801 + 0.996i)19-s + (−0.374 + 0.216i)21-s + (0.581 − 1.00i)23-s + (0.498 − 0.863i)25-s + 0.192i·27-s + (−0.267 − 0.154i)29-s + 0.380i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.940 - 0.338i$
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: \((2,\ 912,\ (\ :1),\ 0.940 - 0.338i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.691055477\)
\(L(\frac12)\) \(\approx\) \(1.691055477\)
\(L(2)\) 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 \( 1 + (1.52 - 18.9i)T \)
good5 \( 1 + (0.140 + 0.243i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 5.24T + 49T^{2} \)
11 \( 1 - 1.15T + 121T^{2} \)
13 \( 1 + (6.32 + 3.65i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-7.52 - 13.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-13.3 + 23.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (7.76 + 4.48i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 11.7iT - 961T^{2} \)
37 \( 1 + 36.1iT - 1.36e3T^{2} \)
41 \( 1 + (-40.3 + 23.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (1.64 + 2.84i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (26.3 - 45.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-88.0 - 50.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-39.8 + 22.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (50.7 - 87.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-50.9 - 29.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-90.7 + 52.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-63.1 - 109. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-55.9 + 32.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 22.0T + 6.88e3T^{2} \)
89 \( 1 + (-107. - 62.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-162. + 93.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.20445955548528017351044204431, −9.091737091381531183874191276709, −8.254551523403287182617737894833, −7.47175368340975909610298151500, −6.37637958156812020801121269759, −5.53864960146803408653538869225, −4.65796602419369233951996435017, −3.77668096921740774536126993162, −2.32253493085447466606680966958, −0.909853776131666780139006887487, 0.831826667802921833506402331289, 2.10792633475896101142864991824, 3.42517744349326854466079938696, 4.87560118490623120884793580106, 5.22202268517950118144557482222, 6.54589840299796472887434236439, 7.27063871252712210566822599948, 8.001935310903288943345245645801, 9.118716239608294735298302561362, 9.780191211361642059565202317469

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