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Properties

Label 2-450-45.34-c1-0-4
Degree $2$
Conductor $450$
Sign $0.114 - 0.993i$
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 − 1.73·3-s + (0.499 + 0.866i)4-s + (−1.49 − 0.866i)6-s + (1.73 + i)7-s + 0.999i·8-s + 2.99·9-s + (−0.866 − 1.49i)12-s + (−3.46 + 2i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s + 6i·17-s + (2.59 + 1.49i)18-s + 7·19-s + (−2.99 − 1.73i)21-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s − 1.00·3-s + (0.249 + 0.433i)4-s + (−0.612 − 0.353i)6-s + (0.654 + 0.377i)7-s + 0.353i·8-s + 0.999·9-s + (−0.249 − 0.433i)12-s + (−0.960 + 0.554i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45i·17-s + (0.612 + 0.353i)18-s + 1.60·19-s + (−0.654 − 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.114 - 0.993i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09403 + 0.975350i\)
\(L(\frac12)\) \(\approx\) \(1.09403 + 0.975350i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
5 \( 1 \)
good7 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 + 6.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-14.7 - 8.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.48075955657037330139332059220, −10.64070758111687116433295551254, −9.633986638589613777172227969799, −8.394267202236720998223635682194, −7.33564427438749264516136564431, −6.56616372723269755128472040618, −5.37999641563612276641408699400, −4.93061296537513794508308742642, −3.62396070854871163622441942319, −1.77510522188239312836898970310, 0.923050122212285611337908036899, 2.70140185017344583631982108111, 4.29754635849590239302748258023, 5.06874552614248846357706697687, 5.84966417172568131262316635849, 7.17675994840916348818904059201, 7.73600425056645330039520316328, 9.618969000123974558111537401738, 10.03456794960830680652036919548, 11.30421990991975353657058862508

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