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Properties

Label 2-882-63.20-c1-0-5
Degree 22
Conductor 882882
Sign 0.748−0.663i0.748 - 0.663i
Analytic cond. 7.042807.04280
Root an. cond. 2.653822.65382
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(882s/2 ΓC(s) L(s)=((0.748−0.663i) Λ‾(2−s)\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}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((0.748−0.663i) Λ‾(1−s)\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: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.748−0.663i0.748 - 0.663i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(587,⋅)\chi_{882} (587, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), 0.748−0.663i)(2,\ 882,\ (\ :1/2),\ 0.748 - 0.663i)

Particular Values

L(1)L(1) ≈\approx 0.731475+0.277494i0.731475 + 0.277494i
L(12)L(\frac12) ≈\approx 0.731475+0.277494i0.731475 + 0.277494i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(1.62−0.601i)T 1 + (1.62 - 0.601i)T
7 1 1
good5 1+(−0.0338−0.0585i)T+(−2.5+4.33i)T2 1 + (-0.0338 - 0.0585i)T + (-2.5 + 4.33i)T^{2}
11 1+(−3.40−1.96i)T+(5.5+9.52i)T2 1 + (-3.40 - 1.96i)T + (5.5 + 9.52i)T^{2}
13 1+(3.32−1.92i)T+(6.5−11.2i)T2 1 + (3.32 - 1.92i)T + (6.5 - 11.2i)T^{2}
17 1−1.55T+17T2 1 - 1.55T + 17T^{2}
19 1+5.84iT−19T2 1 + 5.84iT - 19T^{2}
23 1+(4.78−2.76i)T+(11.5−19.9i)T2 1 + (4.78 - 2.76i)T + (11.5 - 19.9i)T^{2}
29 1+(−1.20−0.697i)T+(14.5+25.1i)T2 1 + (-1.20 - 0.697i)T + (14.5 + 25.1i)T^{2}
31 1+(1.09−0.632i)T+(15.5−26.8i)T2 1 + (1.09 - 0.632i)T + (15.5 - 26.8i)T^{2}
37 1−8.71T+37T2 1 - 8.71T + 37T^{2}
41 1+(−5.17−8.96i)T+(−20.5+35.5i)T2 1 + (-5.17 - 8.96i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.735+1.27i)T+(−21.5−37.2i)T2 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.77−3.06i)T+(−23.5−40.7i)T2 1 + (1.77 - 3.06i)T + (-23.5 - 40.7i)T^{2}
53 1−7.26iT−53T2 1 - 7.26iT - 53T^{2}
59 1+(−4.70−8.14i)T+(−29.5+51.0i)T2 1 + (-4.70 - 8.14i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.0705+0.0407i)T+(30.5+52.8i)T2 1 + (0.0705 + 0.0407i)T + (30.5 + 52.8i)T^{2}
67 1+(−7.67−13.2i)T+(−33.5+58.0i)T2 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2}
71 1−4.30iT−71T2 1 - 4.30iT - 71T^{2}
73 1−7.07iT−73T2 1 - 7.07iT - 73T^{2}
79 1+(−3.42+5.92i)T+(−39.5−68.4i)T2 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.93−6.81i)T+(−41.5−71.8i)T2 1 + (3.93 - 6.81i)T + (-41.5 - 71.8i)T^{2}
89 1−11.6T+89T2 1 - 11.6T + 89T^{2}
97 1+(−0.363−0.209i)T+(48.5+84.0i)T2 1 + (-0.363 - 0.209i)T + (48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line