↑

Properties

Label 2-882-441.130-c1-0-53
Degree $2$
Conductor $882$
Sign $-0.999 - 0.0395i$
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  + (0.826 − 0.563i)2-s + (0.614 − 1.61i)3-s + (0.365 − 0.930i)4-s + (−0.185 − 0.811i)5-s + (−0.404 − 1.68i)6-s + (−2.13 + 1.56i)7-s + (−0.222 − 0.974i)8-s + (−2.24 − 1.98i)9-s + (−0.610 − 0.566i)10-s + (−4.51 − 2.17i)11-s + (−1.28 − 1.16i)12-s + (0.656 − 0.447i)13-s + (−0.876 + 2.49i)14-s + (−1.42 − 0.198i)15-s + (−0.733 − 0.680i)16-s + (1.82 + 4.63i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (0.354 − 0.934i)3-s + (0.182 − 0.465i)4-s + (−0.0828 − 0.362i)5-s + (−0.165 − 0.687i)6-s + (−0.805 + 0.593i)7-s + (−0.0786 − 0.344i)8-s + (−0.748 − 0.663i)9-s + (−0.192 − 0.179i)10-s + (−1.36 − 0.655i)11-s + (−0.370 − 0.335i)12-s + (0.182 − 0.124i)13-s + (−0.234 + 0.667i)14-s + (−0.368 − 0.0512i)15-s + (−0.183 − 0.170i)16-s + (0.441 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 - 0.0395i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.999 - 0.0395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0282714 + 1.42791i\)
\(L(\frac12)\) \(\approx\) \(0.0282714 + 1.42791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.614 + 1.61i)T \)
7 \( 1 + (2.13 - 1.56i)T \)
good5 \( 1 + (0.185 + 0.811i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (4.51 + 2.17i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.656 + 0.447i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-1.82 - 4.63i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.910 + 1.14i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (6.11 + 0.921i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (3.39 + 5.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.13 - 0.925i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-5.75 - 1.77i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-6.91 + 2.13i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-5.47 + 3.73i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-0.215 + 0.0325i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (0.738 - 0.227i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (4.66 + 11.8i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-8.16 - 14.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.19 - 2.75i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.538 - 7.18i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (1.63 - 2.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.90 + 6.75i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (8.34 + 5.68i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-3.01 - 5.23i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.673496632070139657897025339605, −8.767299347649091866184528793345, −8.097691093819610317736514133425, −7.07545835452517994944542198703, −5.96611627289275057280038593654, −5.61036174226742298694571503375, −4.10141263119779379667167423434, −2.93465726031669082465252746403, −2.25651896784554853492147845771, −0.50174401519003271433326808472, 2.53181138170648035265543633101, 3.38575995041016219078081927399, 4.26200057876699669092322223842, 5.22676363887197019163939563817, 6.05518794844096265975142520820, 7.37634149233668721982595221690, 7.69424019559825176311083127154, 9.010341847572682938588016396700, 9.759602907824601333099941844518, 10.59072961273429739222118968610

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