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Properties

Label 2-882-441.121-c1-0-2
Degree $2$
Conductor $882$
Sign $-0.512 - 0.858i$
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  + (−0.900 + 0.433i)2-s + (0.188 − 1.72i)3-s + (0.623 − 0.781i)4-s + (−0.666 + 0.205i)5-s + (0.577 + 1.63i)6-s + (−2.48 − 0.908i)7-s + (−0.222 + 0.974i)8-s + (−2.92 − 0.648i)9-s + (0.511 − 0.474i)10-s + (0.361 − 4.81i)11-s + (−1.22 − 1.22i)12-s + (−0.329 + 4.40i)13-s + (2.63 − 0.259i)14-s + (0.228 + 1.18i)15-s + (−0.222 − 0.974i)16-s + (−2.42 + 6.18i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.108 − 0.994i)3-s + (0.311 − 0.390i)4-s + (−0.297 + 0.0919i)5-s + (0.235 + 0.666i)6-s + (−0.939 − 0.343i)7-s + (−0.0786 + 0.344i)8-s + (−0.976 − 0.216i)9-s + (0.161 − 0.149i)10-s + (0.108 − 1.45i)11-s + (−0.354 − 0.352i)12-s + (−0.0915 + 1.22i)13-s + (0.703 − 0.0694i)14-s + (0.0589 + 0.306i)15-s + (−0.0556 − 0.243i)16-s + (−0.588 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.512 - 0.858i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0592018 + 0.104296i\)
\(L(\frac12)\) \(\approx\) \(0.0592018 + 0.104296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (-0.188 + 1.72i)T \)
7 \( 1 + (2.48 + 0.908i)T \)
good5 \( 1 + (0.666 - 0.205i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.361 + 4.81i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (0.329 - 4.40i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (2.42 - 6.18i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-3.44 + 5.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 0.283i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (3.21 - 8.19i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 - 2.98T + 31T^{2} \)
37 \( 1 + (-5.16 + 0.779i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (4.88 + 4.52i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (-2.18 + 2.02i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (11.8 - 5.71i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.91 - 0.440i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (-1.17 - 5.15i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.73 - 5.93i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 3.04T + 67T^{2} \)
71 \( 1 + (8.18 - 10.2i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.370 + 4.94i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 + (0.343 + 4.58i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-7.27 + 4.96i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (-5.19 - 8.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44237686278637205901588818605, −9.160291895088503456740484601610, −8.830326152325683999181358897532, −7.83925415381786509118768383154, −6.94692196894010767958789781360, −6.45089436453367899373287262917, −5.61358717387813414376424198812, −3.89013616069782818878397097506, −2.84565718426044483637681432071, −1.39430243745984676648679502055, 0.07020647068282025985558331523, 2.35252563641781523210161453063, 3.29453284410133311557871277335, 4.29513061592015958056520410561, 5.36654132960455866371754598768, 6.41385026703844711483585709490, 7.64390136802431075684929219672, 8.201673229421593168639724286934, 9.492842574701182576808380845676, 9.790597158667746826052358022332

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