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Properties

Label 2-442-221.16-c1-0-4
Degree $2$
Conductor $442$
Sign $-0.998 - 0.0531i$
Analytic cond. $3.52938$
Root an. cond. $1.87866$
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 + (−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

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

Invariants

Degree: \(2\)
Conductor: \(442\)    =    \(2 \cdot 13 \cdot 17\)
Sign: $-0.998 - 0.0531i$
Analytic conductor: \(3.52938\)
Root analytic conductor: \(1.87866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{442} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 442,\ (\ :1/2),\ -0.998 - 0.0531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0224242 + 0.843824i\)
\(L(\frac12)\) \(\approx\) \(0.0224242 + 0.843824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.99 + 2.01i)T \)
17 \( 1 + (2.52 + 3.26i)T \)
good3 \( 1 + (0.340 - 0.196i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.77iT - 5T^{2} \)
7 \( 1 + (2.18 + 1.26i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.98 - 1.72i)T + (5.5 - 9.52i)T^{2} \)
19 \( 1 + (3.66 - 6.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.147 - 0.0854i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.57 - 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.12iT - 31T^{2} \)
37 \( 1 + (-9.10 + 5.25i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.111 - 0.0644i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.65 - 4.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.05T + 47T^{2} \)
53 \( 1 + 4.26T + 53T^{2} \)
59 \( 1 + (-0.0723 + 0.125i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-13.4 - 7.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 - 7.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.14 - 2.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.40iT - 73T^{2} \)
79 \( 1 + 2.76iT - 79T^{2} \)
83 \( 1 - 6.67T + 83T^{2} \)
89 \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.50 - 3.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−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 $Z$-function along the critical line