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Properties

Label 2-912-19.12-c2-0-22
Degree $2$
Conductor $912$
Sign $0.962 + 0.271i$
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 + (1.07 + 1.86i)5-s + 5.32·7-s + (1.5 − 2.59i)9-s + 18.2·11-s + (−9.90 − 5.71i)13-s + (−3.22 − 1.86i)15-s + (−8.69 − 15.0i)17-s + (12.2 − 14.5i)19-s + (−7.98 + 4.61i)21-s + (−4.73 + 8.20i)23-s + (10.1 − 17.6i)25-s + 5.19i·27-s + (−8.05 − 4.65i)29-s + 10.1i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.215 + 0.372i)5-s + 0.760·7-s + (0.166 − 0.288i)9-s + 1.65·11-s + (−0.762 − 0.439i)13-s + (−0.215 − 0.124i)15-s + (−0.511 − 0.886i)17-s + (0.644 − 0.764i)19-s + (−0.380 + 0.219i)21-s + (−0.206 + 0.356i)23-s + (0.407 − 0.705i)25-s + 0.192i·27-s + (−0.277 − 0.160i)29-s + 0.326i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.962 + 0.271i$
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: \((2,\ 912,\ (\ :1),\ 0.962 + 0.271i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.883277337\)
\(L(\frac12)\) \(\approx\) \(1.883277337\)
\(L(2)\) 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 \( 1 + (-12.2 + 14.5i)T \)
good5 \( 1 + (-1.07 - 1.86i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 5.32T + 49T^{2} \)
11 \( 1 - 18.2T + 121T^{2} \)
13 \( 1 + (9.90 + 5.71i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (8.69 + 15.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (4.73 - 8.20i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (8.05 + 4.65i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 10.1iT - 961T^{2} \)
37 \( 1 - 5.66iT - 1.36e3T^{2} \)
41 \( 1 + (1.13 - 0.652i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.23 + 14.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-11.7 + 20.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-9.79 - 5.65i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-65.2 + 37.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.9 + 79.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.19 + 5.31i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-102. + 59.4i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-70.7 - 122. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (58.0 - 33.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 135.T + 6.88e3T^{2} \)
89 \( 1 + (68.1 + 39.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-41.2 + 23.7i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.770737345032591726718173394991, −9.252630211002766191755794508702, −8.208755500790750567641673945888, −7.06091523297615509427935002728, −6.57274190038766906802896590467, −5.33670116251938534955910065335, −4.66899637579216873533529255335, −3.56289468719413444912405509143, −2.22090738028388846626393438151, −0.78540236741181396731044558320, 1.15646027389919714450008451813, 1.99132307531973241936439577955, 3.79400049194202982671351710433, 4.64125765228977650967118244234, 5.60361306945749457440629757034, 6.49451312803682091894009728750, 7.28969963224235609069800882539, 8.288235520796197035692815321346, 9.113545624698024078574885596128, 9.861163974318349460163860752938

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