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Properties

Label 2-450-15.8-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.374 - 0.927i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.374 - 0.927i$
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: \((2,\ 450,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867362 + 1.28547i\)
\(L(\frac12)\) \(\approx\) \(0.867362 + 1.28547i\)
\(L(\frac{3}{2})\) 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

−11.64391103104480131664110395668, −10.44930049945456477171009077035, −9.373469719333645438184249215146, −8.716928959837207772650199635794, −7.53476956610735752502188404998, −6.54964679764010599112648096555, −5.87133931367435256965919071251, −4.64264949554590850888125351361, −3.58731091892657311913817417962, −2.19325509155823993485825402202, 0.852917136872713109943982742994, 2.85474579952189291429070302072, 3.70381164585926326203670811295, 4.88818156063063293816677009933, 6.11648095784285836951446656422, 6.82742607492512734732687657182, 8.171731286314298625227412212456, 9.147488110910571738775811353048, 10.11738057473492511181494849375, 11.01741655733978773939803375836

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