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Properties

Label 2-450-45.2-c1-0-1
Degree $2$
Conductor $450$
Sign $-0.882 + 0.469i$
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 + (−0.0795 + 1.73i)3-s + (−0.866 + 0.499i)4-s + (−1.69 + 0.370i)6-s + (−2.32 + 0.622i)7-s + (−0.707 − 0.707i)8-s + (−2.98 − 0.275i)9-s + (0.991 + 0.572i)11-s + (−0.796 − 1.53i)12-s + (−2.38 − 0.640i)13-s + (−1.20 − 2.08i)14-s + (0.500 − 0.866i)16-s + (−4.99 + 4.99i)17-s + (−0.507 − 2.95i)18-s + 2.78i·19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.0459 + 0.998i)3-s + (−0.433 + 0.249i)4-s + (−0.690 + 0.151i)6-s + (−0.877 + 0.235i)7-s + (−0.249 − 0.249i)8-s + (−0.995 − 0.0917i)9-s + (0.299 + 0.172i)11-s + (−0.229 − 0.444i)12-s + (−0.662 − 0.177i)13-s + (−0.321 − 0.556i)14-s + (0.125 − 0.216i)16-s + (−1.21 + 1.21i)17-s + (−0.119 − 0.696i)18-s + 0.638i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.882 + 0.469i$
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: \((2,\ 450,\ (\ :1/2),\ -0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186224 - 0.746402i\)
\(L(\frac12)\) \(\approx\) \(0.186224 - 0.746402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0795 - 1.73i)T \)
5 \( 1 \)
good7 \( 1 + (2.32 - 0.622i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.991 - 0.572i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.38 + 0.640i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.99 - 4.99i)T - 17iT^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (-1.59 + 5.95i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.672 - 1.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.16 - 8.16i)T + 37iT^{2} \)
41 \( 1 + (1.70 - 0.986i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 8.68i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.19 + 11.9i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.84 - 1.84i)T + 53iT^{2} \)
59 \( 1 + (1.31 + 2.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.54 - 6.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0146 + 0.0545i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 + (7.82 - 7.82i)T - 73iT^{2} \)
79 \( 1 + (-8.46 - 4.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.70 - 0.724i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.87T + 89T^{2} \)
97 \( 1 + (-7.79 + 2.08i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.58402818270668311988491788470, −10.44289099006119476865714773488, −9.798689878486600200852570602124, −8.883433486267924060511842744388, −8.157772794261504640183126216590, −6.66808290726823159032055282169, −6.08051892338955564570654734183, −4.84977400980565249502760536360, −4.01340450145995197058390443903, −2.79237840203158555000141916953, 0.43364156881914278035876237749, 2.20371032966187887276499677687, 3.21660848370400612470078953024, 4.65886130099684316826381546879, 5.90777553042845007333794021597, 6.88121415572596846664177213499, 7.63501030862096130970842030100, 9.128232143792033020474713832902, 9.460230938494632934554436015260, 10.89717469094208575807548976135

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