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Properties

Label 2-450-45.4-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.972 + 0.232i$
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 + (0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + (−1.5 − 0.866i)6-s + (−3.46 + 2i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 − 2.59i)11-s + 1.73·12-s + (−3.46 − 2i)13-s + (1.99 − 3.46i)14-s + (−0.5 − 0.866i)16-s + 3i·17-s − 3i·18-s − 5·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + (−0.612 − 0.353i)6-s + (−1.30 + 0.755i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.452 − 0.783i)11-s + 0.499·12-s + (−0.960 − 0.554i)13-s + (0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.727i·17-s − 0.707i·18-s − 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.972 + 0.232i$
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: \((2,\ 450,\ (\ :1/2),\ -0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0560081 - 0.475523i\)
\(L(\frac12)\) \(\approx\) \(0.0560081 - 0.475523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.46 + 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.52 - 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 - 6i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-9.52 + 5.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.18386031140838824199255064688, −10.37491437473307302732893728906, −9.729185973000450787578992423281, −8.846117277597089611596954835546, −8.308013010848122820228918670237, −7.03008774327176204295031878194, −5.91878906940466931202545455654, −5.07254469367324186838827404976, −3.44074413769575506487357579105, −2.54908149512236029389980213188, 0.31407741482225427613906979731, 2.18884700741170216058867990725, 3.14805546085180496860853721144, 4.54689065441037875179169204744, 6.47936131588482865718397035950, 6.97582969063527856675569584815, 7.78371973550603916365600195764, 8.913333416462490779300503180634, 9.709748718331895682224511132547, 10.33627718147698500806231928720

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