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Properties

Label 2-450-225.106-c1-0-7
Degree $2$
Conductor $450$
Sign $0.291 - 0.956i$
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  + (−0.978 − 0.207i)2-s + (1.70 + 0.291i)3-s + (0.913 + 0.406i)4-s + (0.742 + 2.10i)5-s + (−1.60 − 0.639i)6-s + (−1.24 + 2.15i)7-s + (−0.809 − 0.587i)8-s + (2.83 + 0.994i)9-s + (−0.288 − 2.21i)10-s + (1.89 + 0.402i)11-s + (1.44 + 0.960i)12-s + (−6.05 + 1.28i)13-s + (1.66 − 1.85i)14-s + (0.654 + 3.81i)15-s + (0.669 + 0.743i)16-s + (2.42 + 1.76i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (0.985 + 0.168i)3-s + (0.456 + 0.203i)4-s + (0.332 + 0.943i)5-s + (−0.657 − 0.261i)6-s + (−0.471 + 0.816i)7-s + (−0.286 − 0.207i)8-s + (0.943 + 0.331i)9-s + (−0.0911 − 0.701i)10-s + (0.570 + 0.121i)11-s + (0.416 + 0.277i)12-s + (−1.67 + 0.356i)13-s + (0.445 − 0.495i)14-s + (0.168 + 0.985i)15-s + (0.167 + 0.185i)16-s + (0.589 + 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.291 - 0.956i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.291 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09856 + 0.813690i\)
\(L(\frac12)\) \(\approx\) \(1.09856 + 0.813690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (-1.70 - 0.291i)T \)
5 \( 1 + (-0.742 - 2.10i)T \)
good7 \( 1 + (1.24 - 2.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.89 - 0.402i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (6.05 - 1.28i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-2.42 - 1.76i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.08 + 1.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.874 + 0.970i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (0.809 + 7.69i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (0.638 - 6.07i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-1.73 + 5.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-9.61 + 2.04i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-5.32 + 9.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 11.0i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (1.58 - 1.15i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-8.07 + 1.71i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-4.08 - 0.868i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.787 - 7.49i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (-2.33 + 1.69i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.58 - 7.94i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.925 + 8.80i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-4.82 + 2.14i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.26 + 6.98i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.32 + 12.6i)T + (-94.8 + 20.1i)T^{2} \)
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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

−11.01446915456753812316547157733, −10.03976977764814736935331717802, −9.513393340188363112527173387969, −8.824222077733647263707510000188, −7.59817652181320157991728049243, −6.98304617695785713145156231513, −5.84168312778793729872345526795, −4.14244120362179232878181664955, −2.78713147479978627046622746261, −2.18705292957401702760725784797, 0.987403052884483981941329430605, 2.46369065342888766526790490166, 3.88186860997465053574428763604, 5.12545265158606118076850524236, 6.57524057073089526448177798069, 7.49716449285759419943205439494, 8.143866705296165454258136539159, 9.358565424190242461687926260750, 9.590860675551163097832298208463, 10.48636831888808002472048786484

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