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Properties

Label 2-912-19.8-c2-0-1
Degree $2$
Conductor $912$
Sign $-0.946 + 0.321i$
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 + (−3 + 5.19i)5-s + 5·7-s + (1.5 + 2.59i)9-s + (−16.5 + 9.52i)13-s + (9 − 5.19i)15-s + (−3 + 5.19i)17-s + (13 + 13.8i)19-s + (−7.5 − 4.33i)21-s + (−12 − 20.7i)23-s + (−5.5 − 9.52i)25-s − 5.19i·27-s + (27 − 15.5i)29-s − 29.4i·31-s + (−15 + 25.9i)35-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.600 + 1.03i)5-s + 0.714·7-s + (0.166 + 0.288i)9-s + (−1.26 + 0.732i)13-s + (0.599 − 0.346i)15-s + (−0.176 + 0.305i)17-s + (0.684 + 0.729i)19-s + (−0.357 − 0.206i)21-s + (−0.521 − 0.903i)23-s + (−0.220 − 0.381i)25-s − 0.192i·27-s + (0.931 − 0.537i)29-s − 0.949i·31-s + (−0.428 + 0.742i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.946 + 0.321i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ -0.946 + 0.321i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1542491974\)
\(L(\frac12)\) \(\approx\) \(0.1542491974\)
\(L(2)\) 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 \( 1 + (-13 - 13.8i)T \)
good5 \( 1 + (3 - 5.19i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + (16.5 - 9.52i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (12 + 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-27 + 15.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 29.4iT - 961T^{2} \)
37 \( 1 - 60.6iT - 1.36e3T^{2} \)
41 \( 1 + (36 + 20.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (12.5 - 21.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (21 + 36.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-54 + 31.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (63 + 36.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (21.5 + 37.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (49.5 - 28.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (54 + 31.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 126T + 6.88e3T^{2} \)
89 \( 1 + (9 - 5.19i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (114 + 65.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.32665792727635543057324546143, −9.854031842611033756411292585747, −8.408258494524091025317623760409, −7.73657486242439799146546842057, −6.95128026036942722069777889816, −6.26513088224085724924564731362, −5.02324834279201588034082146053, −4.21938234099063058897154533427, −2.92004431565720049781975229487, −1.76791714782123571490302283353, 0.05610037965104912722668588883, 1.29637605717862797527193283291, 2.95630584492293215192943945633, 4.34712322953427252703238614140, 4.96793362982337263788817556352, 5.57259733977194787694948563650, 7.06158990459906161662017566222, 7.73774751553201633396306582281, 8.622889102468590100163978499284, 9.402694254736172943927003576163

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