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Properties

Label 2-882-63.4-c1-0-6
Degree $2$
Conductor $882$
Sign $0.913 - 0.406i$
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  + (0.5 − 0.866i)2-s + (−1.29 − 1.15i)3-s + (−0.499 − 0.866i)4-s − 1.58·5-s + (−1.64 + 0.545i)6-s − 0.999·8-s + (0.349 + 2.97i)9-s + (−0.794 + 1.37i)10-s − 1.58·11-s + (−0.349 + 1.69i)12-s + (−2.40 + 4.16i)13-s + (2.05 + 1.82i)15-s + (−0.5 + 0.866i)16-s + (2.69 − 4.67i)17-s + (2.75 + 1.18i)18-s + (3.54 + 6.14i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.747 − 0.664i)3-s + (−0.249 − 0.433i)4-s − 0.710·5-s + (−0.671 + 0.222i)6-s − 0.353·8-s + (0.116 + 0.993i)9-s + (−0.251 + 0.434i)10-s − 0.478·11-s + (−0.100 + 0.489i)12-s + (−0.667 + 1.15i)13-s + (0.530 + 0.472i)15-s + (−0.125 + 0.216i)16-s + (0.654 − 1.13i)17-s + (0.649 + 0.279i)18-s + (0.814 + 1.41i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709465 + 0.150532i\)
\(L(\frac12)\) \(\approx\) \(0.709465 + 0.150532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.29 + 1.15i)T \)
7 \( 1 \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + (2.40 - 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.54 - 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.300T + 23T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.35 + 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.23 - 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.23 - 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (8.02 - 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.712 + 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27954125568838833984507368521, −9.683379554791742061638339236806, −8.385464581888918603436625129913, −7.48488425056266081263394280528, −6.85197870417964930967714364317, −5.60204393742419137039065841598, −4.95078751094005910501521419290, −3.86439888202851027259706602745, −2.59332647475873462420720403912, −1.26516580058419491578603581842, 0.37829664396530860237396457369, 2.98124494861227508944443258946, 3.93041284134526325081779523297, 4.92406378904736415913657689935, 5.54006366560514180036861991828, 6.50731690598978669440943841824, 7.55917851659143289386169401320, 8.139647979495087091602436133704, 9.296059909870284616264994387246, 10.15826934200336078441991665630

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