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Properties

Label 2-882-441.131-c1-0-43
Degree $2$
Conductor $882$
Sign $0.970 - 0.242i$
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  + (0.781 + 0.623i)2-s + (1.55 + 0.759i)3-s + (0.222 + 0.974i)4-s + (−0.205 − 2.74i)5-s + (0.743 + 1.56i)6-s + (−2.41 − 1.07i)7-s + (−0.433 + 0.900i)8-s + (1.84 + 2.36i)9-s + (1.54 − 2.27i)10-s + (4.21 − 1.65i)11-s + (−0.394 + 1.68i)12-s + (4.33 − 1.70i)13-s + (−1.21 − 2.34i)14-s + (1.76 − 4.42i)15-s + (−0.900 + 0.433i)16-s + (−0.785 + 0.242i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.898 + 0.438i)3-s + (0.111 + 0.487i)4-s + (−0.0919 − 1.22i)5-s + (0.303 + 0.638i)6-s + (−0.913 − 0.406i)7-s + (−0.153 + 0.318i)8-s + (0.615 + 0.788i)9-s + (0.490 − 0.718i)10-s + (1.27 − 0.499i)11-s + (−0.113 + 0.486i)12-s + (1.20 − 0.472i)13-s + (−0.325 − 0.627i)14-s + (0.455 − 1.14i)15-s + (−0.225 + 0.108i)16-s + (−0.190 + 0.0587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.83209 + 0.349014i\)
\(L(\frac12)\) \(\approx\) \(2.83209 + 0.349014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
3 \( 1 + (-1.55 - 0.759i)T \)
7 \( 1 + (2.41 + 1.07i)T \)
good5 \( 1 + (0.205 + 2.74i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-4.21 + 1.65i)T + (8.06 - 7.48i)T^{2} \)
13 \( 1 + (-4.33 + 1.70i)T + (9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.785 - 0.242i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.24 - 5.64i)T + (-1.71 + 22.9i)T^{2} \)
29 \( 1 + (1.03 + 3.34i)T + (-23.9 + 16.3i)T^{2} \)
31 \( 1 + 4.41iT - 31T^{2} \)
37 \( 1 + (-4.13 - 3.84i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (9.52 - 6.49i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (2.29 + 1.56i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (-4.99 + 6.25i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.43 - 1.55i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (1.48 - 0.714i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (10.3 + 2.35i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 3.36T + 67T^{2} \)
71 \( 1 + (13.3 - 3.04i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.78 + 3.44i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (2.53 - 6.44i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (10.9 - 1.64i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-7.12 + 4.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.889801131970958526238837850324, −9.014152705797301855159373493700, −8.694886883259839661405727104111, −7.73432271729555884222507359013, −6.66291654843121989702848573904, −5.78392394445416571447319701742, −4.61779175232859646954582774214, −3.85495350911095850077705154177, −3.15968863674172291594015081746, −1.27394753735796888419365587624, 1.59479633764392463648650425735, 2.79684349513705373838091692725, 3.45807281685971406109587711055, 4.29803881823350462078702785540, 6.12853545196022097844117928370, 6.66490552708624783333017742462, 7.17262361826516990284738602708, 8.789627535142318409508422898655, 9.097540916569859394671864905457, 10.22884030646983927355739767046

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