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Properties

Label 2-450-45.2-c1-0-9
Degree $2$
Conductor $450$
Sign $0.0572 + 0.998i$
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 + (−1.67 − 0.448i)3-s + (−0.866 + 0.499i)4-s + 1.73i·6-s + (1.67 − 0.448i)7-s + (0.707 + 0.707i)8-s + (2.59 + 1.50i)9-s + (3 + 1.73i)11-s + (1.67 − 0.448i)12-s + (−3.34 − 0.896i)13-s + (−0.866 − 1.50i)14-s + (0.500 − 0.866i)16-s + (4.24 − 4.24i)17-s + (0.776 − 2.89i)18-s + 2i·19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.965 − 0.258i)3-s + (−0.433 + 0.249i)4-s + 0.707i·6-s + (0.632 − 0.169i)7-s + (0.249 + 0.249i)8-s + (0.866 + 0.5i)9-s + (0.904 + 0.522i)11-s + (0.482 − 0.129i)12-s + (−0.928 − 0.248i)13-s + (−0.231 − 0.400i)14-s + (0.125 − 0.216i)16-s + (1.02 − 1.02i)17-s + (0.183 − 0.683i)18-s + 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0572 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0572 + 0.998i$
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: \((2,\ 450,\ (\ :1/2),\ 0.0572 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.700702 - 0.661665i\)
\(L(\frac12)\) \(\approx\) \(0.700702 - 0.661665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 \)
good7 \( 1 + (-1.67 + 0.448i)T + (6.06 - 3.5i)T^{2} \)
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 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-0.776 + 2.89i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.33 + 7.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + (-7.5 + 4.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.68 - 10.0i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.776 + 2.89i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.13 - 11.7i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \)
79 \( 1 + (3.46 + 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.89 + 0.776i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (6.69 - 1.79i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.08840674001002895805331066834, −9.959166995800565477481830162855, −9.543110468398387735201336334998, −7.953416050173211106216700608329, −7.34155510383516711239581306158, −6.08146731266289102833600101984, −4.95076723372867421573891376044, −4.14657661450539837145790938205, −2.33838067013545834085708997632, −0.888079192990846891432957398764, 1.31183571840784012925948963713, 3.69576239507093513508112946678, 4.89953281537067892200726897546, 5.60253740670384774148216643101, 6.64226875299126816932407717208, 7.43622608632783723016724105619, 8.628831808058340125017777723537, 9.456782157982306422938954103286, 10.45018041493896324197360405717, 11.21362803978024630654725513394

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