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Properties

Label 2-450-45.34-c1-0-12
Degree $2$
Conductor $450$
Sign $0.726 + 0.687i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.726 + 0.687i$
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: \((2,\ 450,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36164 - 0.542346i\)
\(L(\frac12)\) \(\approx\) \(1.36164 - 0.542346i\)
\(L(\frac{3}{2})\) 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

−10.71995123177508279868259284758, −9.924574838540558769600153031774, −9.232738079544981803252193344640, −8.433844935043227387169979268508, −7.42477684527285694488093438504, −6.76794444576115552892732635682, −5.11307092580265411474401431449, −3.51888415117457173888519504380, −2.96885164544055880103375250322, −1.20675279248670343376122428811, 1.59986130675274359312721141226, 3.05150949865388807070070605412, 4.17442594640257296276046123673, 5.85113760994653355153170394460, 6.64907816144280825636784433537, 7.84305900251725705959706272948, 8.398835961302012285798911981180, 9.493757048119653794763901736984, 9.791878898987060058327876881894, 11.03378592969926699014479574190

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