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Properties

Label 2-882-49.37-c1-0-4
Degree $2$
Conductor $882$
Sign $0.664 - 0.747i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−3.05 − 0.943i)5-s + (−1.76 + 1.96i)7-s + (−0.222 − 0.974i)8-s + (−3.05 + 0.943i)10-s + (0.192 + 2.56i)11-s + (6.17 + 2.97i)13-s + (−0.352 + 2.62i)14-s + (−0.733 − 0.680i)16-s + (1.47 + 0.221i)17-s + (−3.35 + 5.80i)19-s + (−1.99 + 2.50i)20-s + (1.60 + 2.01i)22-s + (−1.36 + 0.206i)23-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (0.182 − 0.465i)4-s + (−1.36 − 0.422i)5-s + (−0.668 + 0.743i)7-s + (−0.0786 − 0.344i)8-s + (−0.967 + 0.298i)10-s + (0.0579 + 0.773i)11-s + (1.71 + 0.825i)13-s + (−0.0942 + 0.700i)14-s + (−0.183 − 0.170i)16-s + (0.357 + 0.0538i)17-s + (−0.768 + 1.33i)19-s + (−0.446 + 0.559i)20-s + (0.341 + 0.428i)22-s + (−0.285 + 0.0429i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.664 - 0.747i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.664 - 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21208 + 0.544221i\)
\(L(\frac12)\) \(\approx\) \(1.21208 + 0.544221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
3 \( 1 \)
7 \( 1 + (1.76 - 1.96i)T \)
good5 \( 1 + (3.05 + 0.943i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.192 - 2.56i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-6.17 - 2.97i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-1.47 - 0.221i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (3.35 - 5.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.36 - 0.206i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-5.60 + 7.03i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.922 - 1.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.18 - 8.12i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.09 - 9.16i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.06 + 4.65i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (7.96 - 5.42i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (3.44 - 8.77i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (1.85 - 0.572i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-1.83 - 4.66i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (3.77 + 6.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.630 + 0.790i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (7.37 + 5.02i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-0.795 + 1.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.83 - 4.25i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.05 - 14.1i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 3.99T + 97T^{2} \)
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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

−10.33408682740577040011716407188, −9.476423333518427037635914086548, −8.459914630725342736817749244014, −7.917482931167435978706894725199, −6.49912228776450884243132314219, −6.02190399167630690449359415633, −4.53330286400299700493737127766, −4.01878397957039974695646128742, −3.03001766965223464252352320310, −1.47603042648069071137479677169, 0.56684947010296454199577054956, 3.10076903045794725051015293197, 3.59572879417347357898319095962, 4.44018160855198807496070852508, 5.79446663720188732687117100785, 6.63731595125809257799137264641, 7.32679628008025814375771974227, 8.255637824581008128179744115319, 8.806835048848833507840415374113, 10.33458266092565188338945185769

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