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Properties

Label 2-450-45.32-c1-0-7
Degree $2$
Conductor $450$
Sign $0.999 + 0.0253i$
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  + (0.965 + 0.258i)2-s + (−1.73 + 0.0795i)3-s + (0.866 + 0.499i)4-s + (−1.69 − 0.370i)6-s + (0.622 − 2.32i)7-s + (0.707 + 0.707i)8-s + (2.98 − 0.275i)9-s + (0.991 − 0.572i)11-s + (−1.53 − 0.796i)12-s + (0.640 + 2.38i)13-s + (1.20 − 2.08i)14-s + (0.500 + 0.866i)16-s + (4.99 − 4.99i)17-s + (2.95 + 0.507i)18-s + 2.78i·19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.998 + 0.0459i)3-s + (0.433 + 0.249i)4-s + (−0.690 − 0.151i)6-s + (0.235 − 0.877i)7-s + (0.249 + 0.249i)8-s + (0.995 − 0.0917i)9-s + (0.299 − 0.172i)11-s + (−0.444 − 0.229i)12-s + (0.177 + 0.662i)13-s + (0.321 − 0.556i)14-s + (0.125 + 0.216i)16-s + (1.21 − 1.21i)17-s + (0.696 + 0.119i)18-s + 0.638i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0253i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.999 + 0.0253i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67788 - 0.0212777i\)
\(L(\frac12)\) \(\approx\) \(1.67788 - 0.0212777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (1.73 - 0.0795i)T \)
5 \( 1 \)
good7 \( 1 + (-0.622 + 2.32i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.991 + 0.572i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.640 - 2.38i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.99 + 4.99i)T - 17iT^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (-5.95 + 1.59i)T + (19.9 - 11.5i)T^{2} \)
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 \( 1 + (8.68 + 2.32i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (11.9 + 3.19i)T + (40.7 + 23.5i)T^{2} \)
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 \( 1 + (0.0545 - 0.0146i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (0.724 - 2.70i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (2.08 - 7.79i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.43195283324765763192311426566, −10.35801001572794979422209413351, −9.580034658137359917185797653171, −8.044394571475975340391664260395, −7.07008228434245971450114952411, −6.41993621735143083409488357562, −5.23073721879318241300241779597, −4.50656200668620566172994348533, −3.34512540671351805477860933922, −1.22944591686365643750469560017, 1.43690244896761297025514824742, 3.11171308103910214530365103215, 4.48647027852101655305074700279, 5.46420325578138016396794133191, 6.05428974908783134368387729671, 7.14559452879319630800694914701, 8.269385777443015235486916705448, 9.555330826867220994921720369744, 10.47333068896326679203788312600, 11.29116529410345373257020419881

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