↑

Properties

Label 2-882-63.5-c1-0-29
Degree $2$
Conductor $882$
Sign $0.463 + 0.886i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.149 + 1.72i)3-s − 4-s + (−0.948 − 1.64i)5-s + (−1.72 + 0.149i)6-s − i·8-s + (−2.95 + 0.514i)9-s + (1.64 − 0.948i)10-s + (0.363 + 0.209i)11-s + (−0.149 − 1.72i)12-s + (−1.54 − 0.891i)13-s + (2.69 − 1.88i)15-s + 16-s + (−1.05 − 1.83i)17-s + (−0.514 − 2.95i)18-s + (−5.50 − 3.17i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0861 + 0.996i)3-s − 0.5·4-s + (−0.424 − 0.735i)5-s + (−0.704 + 0.0608i)6-s − 0.353i·8-s + (−0.985 + 0.171i)9-s + (0.519 − 0.300i)10-s + (0.109 + 0.0631i)11-s + (−0.0430 − 0.498i)12-s + (−0.428 − 0.247i)13-s + (0.695 − 0.486i)15-s + 0.250·16-s + (−0.256 − 0.444i)17-s + (−0.121 − 0.696i)18-s + (−1.26 − 0.728i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.463 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385254 - 0.233356i\)
\(L(\frac12)\) \(\approx\) \(0.385254 - 0.233356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.149 - 1.72i)T \)
7 \( 1 \)
good5 \( 1 + (0.948 + 1.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.363 - 0.209i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.54 + 0.891i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.05 + 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.50 + 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.49 - 3.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.57 + 2.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + (-5.29 + 9.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.03 - 6.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.62T + 47T^{2} \)
53 \( 1 + (6.52 - 3.76i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.10T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 1.21T + 67T^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 + (-9.31 + 5.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.61T + 79T^{2} \)
83 \( 1 + (5.98 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.56 + 2.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.0 - 5.82i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.683007754378923524235920447682, −9.196481206126166998589558465864, −8.267867986806736489763432640315, −7.71875136618179851114696005916, −6.37604340320313446920574121941, −5.54475309134783100337799350568, −4.44970887845616680878171846986, −4.16873629070206189328355709684, −2.58413197526248347123811428124, −0.20787914063900336858598869212, 1.63208863723051342780242310117, 2.66407470397591911694406852363, 3.64989569141384029106104165512, 4.81826966268271443446785718860, 6.29936954669138582133873078086, 6.69891797293317511816207744209, 8.010634651963570294616978844701, 8.341687942522669486436398972192, 9.527251613498012531841354079616, 10.55446813605203927534671901544

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