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Properties

Label 2-912-19.12-c2-0-8
Degree $2$
Conductor $912$
Sign $-0.178 - 0.983i$
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 + (2.27 + 3.93i)5-s − 9.87·7-s + (1.5 − 2.59i)9-s + 15.6·11-s + (13.3 + 7.68i)13-s + (−6.81 − 3.93i)15-s + (−12.4 − 21.5i)17-s + (18.4 + 4.63i)19-s + (14.8 − 8.55i)21-s + (−4.15 + 7.19i)23-s + (2.16 − 3.75i)25-s + 5.19i·27-s + (27.3 + 15.7i)29-s + 30.9i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.454 + 0.787i)5-s − 1.41·7-s + (0.166 − 0.288i)9-s + 1.42·11-s + (1.02 + 0.590i)13-s + (−0.454 − 0.262i)15-s + (−0.730 − 1.26i)17-s + (0.969 + 0.243i)19-s + (0.705 − 0.407i)21-s + (−0.180 + 0.312i)23-s + (0.0866 − 0.150i)25-s + 0.192i·27-s + (0.941 + 0.543i)29-s + 0.997i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.178 - 0.983i$
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: \((2,\ 912,\ (\ :1),\ -0.178 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.414078683\)
\(L(\frac12)\) \(\approx\) \(1.414078683\)
\(L(2)\) 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 \( 1 + (-18.4 - 4.63i)T \)
good5 \( 1 + (-2.27 - 3.93i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 9.87T + 49T^{2} \)
11 \( 1 - 15.6T + 121T^{2} \)
13 \( 1 + (-13.3 - 7.68i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (12.4 + 21.5i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (4.15 - 7.19i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-27.3 - 15.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 30.9iT - 961T^{2} \)
37 \( 1 - 17.3iT - 1.36e3T^{2} \)
41 \( 1 + (44.2 - 25.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-0.773 - 1.33i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.09 - 8.81i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.54 - 2.62i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (68.4 - 39.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.9 - 93.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.9 - 9.80i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-45.3 + 26.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (36.6 + 63.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-27.1 + 15.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 42.6T + 6.88e3T^{2} \)
89 \( 1 + (-114. - 66.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (70.2 - 40.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.08447826352450568485352766871, −9.372688490226344343896418966952, −8.851789108186363834916322102917, −7.16244875401460587326474370043, −6.47387822078941677468842741556, −6.22978579470562081838103864847, −4.83643140963550739448835301131, −3.64567599042665001079865738069, −2.93986603321891299560666296814, −1.21485399246161615925280146100, 0.56519641296358415642327890723, 1.67462617869267589072875714496, 3.33117080801100805056320754722, 4.22039141134174538923667544976, 5.52265389006631953987977814102, 6.30736972192497118948462553643, 6.69091219669531619379820343948, 8.096274113746696672771814865294, 8.992069804273901805874715793054, 9.553412847559834028678650622063

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