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Properties

Label 2-450-45.2-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.0165 - 0.999i$
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  + (−0.258 − 0.965i)2-s + (−1.22 + 1.22i)3-s + (−0.866 + 0.499i)4-s + (1.49 + 0.866i)6-s + (0.707 + 0.707i)8-s − 2.99i·9-s + (3 + 1.73i)11-s + (0.448 − 1.67i)12-s + (−3.34 − 0.896i)13-s + (0.500 − 0.866i)16-s + (−4.24 + 4.24i)17-s + (−2.89 + 0.776i)18-s + 5i·19-s + (0.896 − 3.34i)22-s + (−1.55 + 5.79i)23-s − 1.73·24-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.707 + 0.707i)3-s + (−0.433 + 0.249i)4-s + (0.612 + 0.353i)6-s + (0.249 + 0.249i)8-s − 0.999i·9-s + (0.904 + 0.522i)11-s + (0.129 − 0.482i)12-s + (−0.928 − 0.248i)13-s + (0.125 − 0.216i)16-s + (−1.02 + 1.02i)17-s + (−0.683 + 0.183i)18-s + 1.14i·19-s + (0.191 − 0.713i)22-s + (−0.323 + 1.20i)23-s − 0.353·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0165 - 0.999i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.0165 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435903 + 0.443183i\)
\(L(\frac12)\) \(\approx\) \(0.435903 + 0.443183i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.34 + 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + (1.55 - 5.79i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.46 - 6i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.13 + 11.7i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.55 - 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.24 + 8.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (-8.57 + 8.57i)T - 73iT^{2} \)
79 \( 1 + (-12.1 - 7i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.69 + 2.32i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (5.01 - 1.34i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.22193771992552093813714560689, −10.46751566629953240736653103346, −9.657388238314901900772328397818, −9.046845769647911210362780437570, −7.74975526783837418204384652494, −6.54182276860356479655656493006, −5.46474901443518012606615069679, −4.35099462065782488730726045388, −3.54627019806280495330287876399, −1.72655886277577251280162186220, 0.45036925148561599043041517604, 2.35130143128719162156682362890, 4.39004340141691865675385905171, 5.26008195697329170038082945214, 6.52432805676620114174845490764, 6.86956844124388163032280157473, 7.950538236883001005260699157429, 8.964316612371151391622495864160, 9.810278236966412670055973896442, 11.12754961205751660151359540819

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