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Properties

Label 2-450-45.4-c1-0-13
Degree $2$
Conductor $450$
Sign $0.406 + 0.913i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
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 + (1.41 − i)3-s + (0.499 − 0.866i)4-s + (0.724 − 1.57i)6-s + (0.389 − 0.224i)7-s − 0.999i·8-s + (1.00 − 2.82i)9-s + (2.44 + 4.24i)11-s + (−0.158 − 1.72i)12-s + (0.389 + 0.224i)13-s + (0.224 − 0.389i)14-s + (−0.5 − 0.866i)16-s − 4.89i·17-s + (−0.548 − 2.94i)18-s − 7.44·19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.816 − 0.577i)3-s + (0.249 − 0.433i)4-s + (0.295 − 0.642i)6-s + (0.147 − 0.0849i)7-s − 0.353i·8-s + (0.333 − 0.942i)9-s + (0.738 + 1.27i)11-s + (−0.0458 − 0.497i)12-s + (0.107 + 0.0623i)13-s + (0.0600 − 0.104i)14-s + (−0.125 − 0.216i)16-s − 1.18i·17-s + (−0.129 − 0.695i)18-s − 1.70·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14893 - 1.39650i\)
\(L(\frac12)\) \(\approx\) \(2.14893 - 1.39650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
5 \( 1 \)
good7 \( 1 + (-0.389 + 0.224i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.389 - 0.224i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (-2.12 - 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.43 + 5.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.55iT - 53T^{2} \)
59 \( 1 + (2.72 - 4.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.301 - 0.174i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + (-8.34 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.71 - 2.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (7.61 - 4.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11546399189281758558175898594, −9.918114618185545332063688787847, −9.231460734051338649750693111317, −8.174315988342544859197082951166, −7.02762341737638020368383076240, −6.50077322079107699869411674076, −4.87666415536503729289740062123, −3.96457159041005449739731872180, −2.67407650321425084919376402984, −1.55997885681308033660358949425, 2.16337020735023492830108763976, 3.59583320787432508317988168847, 4.18290591960772263574508911061, 5.55557151973751445746148560017, 6.44605692748288343963793268720, 7.71100329850388044243923866260, 8.645129430410929377166458770460, 9.075445770982452276529562319335, 10.67547695296741109331837042287, 10.95342318012605137814596465760

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