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Properties

Label 2-450-15.8-c1-0-1
Degree $2$
Conductor $450$
Sign $0.749 - 0.662i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−2 + 2i)7-s + (0.707 − 0.707i)8-s − 2.82i·11-s + (3 + 3i)13-s + 2.82·14-s − 1.00·16-s + (2.82 + 2.82i)17-s + 8i·19-s + (−2.00 + 2.00i)22-s + (−2.82 + 2.82i)23-s − 4.24i·26-s + (−2.00 − 2.00i)28-s + 1.41·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.755 + 0.755i)7-s + (0.250 − 0.250i)8-s − 0.852i·11-s + (0.832 + 0.832i)13-s + 0.755·14-s − 0.250·16-s + (0.685 + 0.685i)17-s + 1.83i·19-s + (−0.426 + 0.426i)22-s + (−0.589 + 0.589i)23-s − 0.832i·26-s + (−0.377 − 0.377i)28-s + 0.262·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863234 + 0.326646i\)
\(L(\frac12)\) \(\approx\) \(0.863234 + 0.326646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 9.89iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−11.21356174076819844227837148203, −10.17868737930022126627189920583, −9.502732121732211484881862503468, −8.528334011543821046279454273742, −7.900040088311131336248345216159, −6.33851724884756246696817067541, −5.81019293001894629572154717756, −3.98524274007720587470232946450, −3.10896650376675559733235462274, −1.55923134881898839499084522956, 0.73094494779091932465361343953, 2.78405041830968613138904900258, 4.21880239992879403616989233072, 5.39471142094228256199896543573, 6.60313085692569675350594865784, 7.20502253290136575683051667347, 8.208234064418915898062240433745, 9.228076798755276400647569343647, 10.05210899540715346941232536203, 10.66692945534821798667467919505

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