Properties

Label 2-442-221.16-c1-0-4
Degree 22
Conductor 442442
Sign 0.9980.0531i-0.998 - 0.0531i
Analytic cond. 3.529383.52938
Root an. cond. 1.878661.87866
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  + (0.5 + 0.866i)2-s + (−0.340 + 0.196i)3-s + (−0.499 + 0.866i)4-s + 1.77i·5-s + (−0.340 − 0.196i)6-s + (−2.18 − 1.26i)7-s − 0.999·8-s + (−1.42 + 2.46i)9-s + (−1.53 + 0.888i)10-s + (−2.98 + 1.72i)11-s − 0.393i·12-s + (2.99 − 2.01i)13-s − 2.52i·14-s + (−0.349 − 0.604i)15-s + (−0.5 − 0.866i)16-s + (−2.52 − 3.26i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.196 + 0.113i)3-s + (−0.249 + 0.433i)4-s + 0.794i·5-s + (−0.138 − 0.0802i)6-s + (−0.824 − 0.476i)7-s − 0.353·8-s + (−0.474 + 0.821i)9-s + (−0.486 + 0.280i)10-s + (−0.899 + 0.519i)11-s − 0.113i·12-s + (0.830 − 0.557i)13-s − 0.673i·14-s + (−0.0901 − 0.156i)15-s + (−0.125 − 0.216i)16-s + (−0.611 − 0.791i)17-s + ⋯

Functional equation

Λ(s)=(442s/2ΓC(s)L(s)=((0.9980.0531i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0531i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(442s/2ΓC(s+1/2)L(s)=((0.9980.0531i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 442 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 442442    =    213172 \cdot 13 \cdot 17
Sign: 0.9980.0531i-0.998 - 0.0531i
Analytic conductor: 3.529383.52938
Root analytic conductor: 1.878661.87866
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ442(237,)\chi_{442} (237, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 442, ( :1/2), 0.9980.0531i)(2,\ 442,\ (\ :1/2),\ -0.998 - 0.0531i)

Particular Values

L(1)L(1) \approx 0.0224242+0.843824i0.0224242 + 0.843824i
L(12)L(\frac12) \approx 0.0224242+0.843824i0.0224242 + 0.843824i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
13 1+(2.99+2.01i)T 1 + (-2.99 + 2.01i)T
17 1+(2.52+3.26i)T 1 + (2.52 + 3.26i)T
good3 1+(0.3400.196i)T+(1.52.59i)T2 1 + (0.340 - 0.196i)T + (1.5 - 2.59i)T^{2}
5 11.77iT5T2 1 - 1.77iT - 5T^{2}
7 1+(2.18+1.26i)T+(3.5+6.06i)T2 1 + (2.18 + 1.26i)T + (3.5 + 6.06i)T^{2}
11 1+(2.981.72i)T+(5.59.52i)T2 1 + (2.98 - 1.72i)T + (5.5 - 9.52i)T^{2}
19 1+(3.666.34i)T+(9.516.4i)T2 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.1470.0854i)T+(11.519.9i)T2 1 + (0.147 - 0.0854i)T + (11.5 - 19.9i)T^{2}
29 1+(3.572.06i)T+(14.525.1i)T2 1 + (3.57 - 2.06i)T + (14.5 - 25.1i)T^{2}
31 19.12iT31T2 1 - 9.12iT - 31T^{2}
37 1+(9.10+5.25i)T+(18.532.0i)T2 1 + (-9.10 + 5.25i)T + (18.5 - 32.0i)T^{2}
41 1+(0.1110.0644i)T+(20.535.5i)T2 1 + (0.111 - 0.0644i)T + (20.5 - 35.5i)T^{2}
43 1+(2.654.60i)T+(21.537.2i)T2 1 + (2.65 - 4.60i)T + (-21.5 - 37.2i)T^{2}
47 1+2.05T+47T2 1 + 2.05T + 47T^{2}
53 1+4.26T+53T2 1 + 4.26T + 53T^{2}
59 1+(0.0723+0.125i)T+(29.551.0i)T2 1 + (-0.0723 + 0.125i)T + (-29.5 - 51.0i)T^{2}
61 1+(13.47.76i)T+(30.5+52.8i)T2 1 + (-13.4 - 7.76i)T + (30.5 + 52.8i)T^{2}
67 1+(4.087.07i)T+(33.5+58.0i)T2 1 + (-4.08 - 7.07i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.142.39i)T+(35.5+61.4i)T2 1 + (-4.14 - 2.39i)T + (35.5 + 61.4i)T^{2}
73 19.40iT73T2 1 - 9.40iT - 73T^{2}
79 1+2.76iT79T2 1 + 2.76iT - 79T^{2}
83 16.67T+83T2 1 - 6.67T + 83T^{2}
89 1+(7.64+13.2i)T+(44.5+77.0i)T2 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.503.17i)T+(48.5+84.0i)T2 1 + (-5.50 - 3.17i)T + (48.5 + 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.38810644098398156068042936950, −10.59673435627475298778336231181, −10.00970033102044973411923513490, −8.614888562860959067991730143197, −7.72231800405374542939796008634, −6.85784007480740348900440405014, −5.98350468552610859010900946814, −4.99916121078982036100515291464, −3.69376169257994322988373393043, −2.61355335306055761772646173358, 0.46032072159158067974973748637, 2.38303816423699569524615034014, 3.61825956219651770364218715273, 4.76071551368952434899738647675, 5.97274792302725122128051225728, 6.48418853584145105322217749038, 8.293985895550713431721496308105, 8.991276916429000272294978795197, 9.677846298890319283776370505052, 11.06115637751582926901872919101

Graph of the ZZ-function along the critical line