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Properties

Label 2-442-221.152-c1-0-20
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: $\chi_{442} (373, \cdot )$
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

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

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