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Properties

Label 2-450-45.38-c1-0-2
Degree $2$
Conductor $450$
Sign $0.309 - 0.950i$
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  + (−0.965 + 0.258i)2-s + (−0.933 − 1.45i)3-s + (0.866 − 0.499i)4-s + (1.27 + 1.16i)6-s + (0.521 + 1.94i)7-s + (−0.707 + 0.707i)8-s + (−1.25 + 2.72i)9-s + (−1.70 − 0.984i)11-s + (−1.53 − 0.796i)12-s + (−1.05 + 3.92i)13-s + (−1.00 − 1.74i)14-s + (0.500 − 0.866i)16-s + (−2.35 − 2.35i)17-s + (0.507 − 2.95i)18-s + 3.70i·19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.539 − 0.842i)3-s + (0.433 − 0.249i)4-s + (0.522 + 0.476i)6-s + (0.197 + 0.736i)7-s + (−0.249 + 0.249i)8-s + (−0.418 + 0.908i)9-s + (−0.514 − 0.296i)11-s + (−0.444 − 0.229i)12-s + (−0.291 + 1.08i)13-s + (−0.269 − 0.466i)14-s + (0.125 − 0.216i)16-s + (−0.572 − 0.572i)17-s + (0.119 − 0.696i)18-s + 0.850i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494670 + 0.359279i\)
\(L(\frac12)\) \(\approx\) \(0.494670 + 0.359279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.933 + 1.45i)T \)
5 \( 1 \)
good7 \( 1 + (-0.521 - 1.94i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.70 + 0.984i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.05 - 3.92i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.35 + 2.35i)T + 17iT^{2} \)
19 \( 1 - 3.70iT - 19T^{2} \)
23 \( 1 + (-6.05 - 1.62i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.74 - 6.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.48 - 6.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
41 \( 1 + (-6.13 + 3.54i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.09 + 2.43i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (7.49 - 2.00i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.03 - 7.03i)T - 53iT^{2} \)
59 \( 1 + (-1.34 - 2.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.37 - 7.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 - 2.19i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 + (1.14 + 1.14i)T + 73iT^{2} \)
79 \( 1 + (10.0 + 5.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.440 - 1.64i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.04T + 89T^{2} \)
97 \( 1 + (2.60 + 9.71i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.24719156262860055740144556910, −10.54961470916138478718858202359, −9.207950017567733065415623266751, −8.607336815097085144005268987836, −7.48898390294917290291177469555, −6.82023693418280794427257539221, −5.77480141961399877675797790397, −4.89794404122782875796199246034, −2.76437945273856861060294169283, −1.52472836765430737792296296879, 0.52002651294977604803428348849, 2.69302983544155910251978175789, 4.06036994330404371449186758591, 5.06966371407456942174790877514, 6.24408292707063801815797542138, 7.36099278234903115893256829606, 8.262399717469242147988791878996, 9.394554341961489812486149138757, 10.03409057744600238462040431522, 11.01498010247111224223355216682

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