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Properties

Label 2-882-63.5-c1-0-11
Degree $2$
Conductor $882$
Sign $-0.0211 - 0.999i$
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  + i·2-s + (1.08 + 1.35i)3-s − 4-s + (−1.77 − 3.07i)5-s + (−1.35 + 1.08i)6-s − i·8-s + (−0.645 + 2.92i)9-s + (3.07 − 1.77i)10-s + (2.61 + 1.51i)11-s + (−1.08 − 1.35i)12-s + (0.888 + 0.513i)13-s + (2.22 − 5.73i)15-s + 16-s + (0.809 + 1.40i)17-s + (−2.92 − 0.645i)18-s + (7.12 + 4.11i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.626 + 0.779i)3-s − 0.5·4-s + (−0.794 − 1.37i)5-s + (−0.551 + 0.442i)6-s − 0.353i·8-s + (−0.215 + 0.976i)9-s + (0.972 − 0.561i)10-s + (0.789 + 0.455i)11-s + (−0.313 − 0.389i)12-s + (0.246 + 0.142i)13-s + (0.574 − 1.48i)15-s + 0.250·16-s + (0.196 + 0.339i)17-s + (−0.690 − 0.152i)18-s + (1.63 + 0.943i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0211 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0211 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0211 - 0.999i$
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: \((2,\ 882,\ (\ :1/2),\ -0.0211 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15985 + 1.18466i\)
\(L(\frac12)\) \(\approx\) \(1.15985 + 1.18466i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 - 1.35i)T \)
7 \( 1 \)
good5 \( 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.61 - 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.888 - 0.513i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.809 - 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.12 - 4.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.90 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.70 - 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98iT - 31T^{2} \)
37 \( 1 + (-2.92 + 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0472 - 0.0817i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.05 - 5.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 + (2.76 - 1.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 - 4.69iT - 61T^{2} \)
67 \( 1 + 0.375T + 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (1.13 - 0.655i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.924T + 79T^{2} \)
83 \( 1 + (5.43 + 9.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.35 + 4.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.979960396506398422101797063462, −9.187526385244773452206821663432, −8.776279629889052965576107999988, −7.895160717948116889371874274291, −7.27270081736117660237873895558, −5.75781319604125841449044317485, −4.94306403675003730814460708725, −4.17462201146594693220407278278, −3.42010502203921448678743187170, −1.32040541936296976377180879051, 0.902533774475721682473288548670, 2.53655264314444122223974117333, 3.26096285656908081611703928950, 3.96560741154988387323104474297, 5.64635272565374313212905569248, 6.80249043236445284616050687554, 7.36269307141032623387783769101, 8.163223937377535639123805008843, 9.200662757126274042726823789387, 9.839623538750473417824755433867

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