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Properties

Label 2-882-63.41-c1-0-17
Degree $2$
Conductor $882$
Sign $0.748 + 0.663i$
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  + (−0.866 + 0.5i)2-s + (−1.62 − 0.601i)3-s + (0.499 − 0.866i)4-s + (0.0338 − 0.0585i)5-s + (1.70 − 0.290i)6-s + 0.999i·8-s + (2.27 + 1.95i)9-s + 0.0676i·10-s + (3.40 − 1.96i)11-s + (−1.33 + 1.10i)12-s + (−3.32 − 1.92i)13-s + (−0.0901 + 0.0747i)15-s + (−0.5 − 0.866i)16-s + 1.55·17-s + (−2.94 − 0.554i)18-s + 5.84i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.937 − 0.347i)3-s + (0.249 − 0.433i)4-s + (0.0151 − 0.0261i)5-s + (0.697 − 0.118i)6-s + 0.353i·8-s + (0.758 + 0.651i)9-s + 0.0213i·10-s + (1.02 − 0.592i)11-s + (−0.384 + 0.319i)12-s + (−0.922 − 0.532i)13-s + (−0.0232 + 0.0193i)15-s + (−0.125 − 0.216i)16-s + 0.376·17-s + (−0.694 − 0.130i)18-s + 1.34i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.748 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731475 - 0.277494i\)
\(L(\frac12)\) \(\approx\) \(0.731475 - 0.277494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (1.62 + 0.601i)T \)
7 \( 1 \)
good5 \( 1 + (-0.0338 + 0.0585i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.40 + 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.32 + 1.92i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 - 5.84iT - 19T^{2} \)
23 \( 1 + (4.78 + 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.20 + 0.697i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.09 + 0.632i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 + (-5.17 + 8.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.735 - 1.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 + 3.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.26iT - 53T^{2} \)
59 \( 1 + (-4.70 + 8.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0705 - 0.0407i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.67 + 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + (-3.42 - 5.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.93 + 6.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-0.363 + 0.209i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04793910710662153433044946421, −9.318487902361295716075717410841, −8.125500234711302261199095767575, −7.55225003712227792159477731001, −6.51768257002029439069686940725, −5.91046178493028391437713425920, −5.02922281179775912801771359249, −3.74391959856946364418454264850, −1.99166799091023044755758029959, −0.65787751492044268084605436967, 1.06448539185662450385962351028, 2.54265174856819648578973628682, 4.09134306417068838801565740465, 4.72804493142342972941012456527, 6.04257344680065628632150827863, 6.85283246896325533468724930418, 7.56017145986780026470190872122, 8.871267198220449236552615738433, 9.654183408536227481265109687921, 10.03447591655597296704247012712

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