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Properties

Label 2-450-5.3-c2-0-7
Degree $2$
Conductor $450$
Sign $0.973 + 0.229i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 2i·4-s + (3 − 3i)7-s + (2 + 2i)8-s − 12·11-s + (12 + 12i)13-s + 6i·14-s − 4·16-s + (12 − 12i)17-s − 20i·19-s + (12 − 12i)22-s + (3 + 3i)23-s − 24·26-s + (−6 − 6i)28-s − 30i·29-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.5i·4-s + (0.428 − 0.428i)7-s + (0.250 + 0.250i)8-s − 1.09·11-s + (0.923 + 0.923i)13-s + 0.428i·14-s − 0.250·16-s + (0.705 − 0.705i)17-s − 1.05i·19-s + (0.545 − 0.545i)22-s + (0.130 + 0.130i)23-s − 0.923·26-s + (−0.214 − 0.214i)28-s − 1.03i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.307696399\)
\(L(\frac12)\) \(\approx\) \(1.307696399\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3 + 3i)T - 49iT^{2} \)
11 \( 1 + 12T + 121T^{2} \)
13 \( 1 + (-12 - 12i)T + 169iT^{2} \)
17 \( 1 + (-12 + 12i)T - 289iT^{2} \)
19 \( 1 + 20iT - 361T^{2} \)
23 \( 1 + (-3 - 3i)T + 529iT^{2} \)
29 \( 1 + 30iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 + (-48 + 48i)T - 1.36e3iT^{2} \)
41 \( 1 - 48T + 1.68e3T^{2} \)
43 \( 1 + (-27 - 27i)T + 1.84e3iT^{2} \)
47 \( 1 + (-27 + 27i)T - 2.20e3iT^{2} \)
53 \( 1 + (12 + 12i)T + 2.80e3iT^{2} \)
59 \( 1 + 60iT - 3.48e3T^{2} \)
61 \( 1 - 32T + 3.72e3T^{2} \)
67 \( 1 + (-3 + 3i)T - 4.48e3iT^{2} \)
71 \( 1 - 48T + 5.04e3T^{2} \)
73 \( 1 + (-12 - 12i)T + 5.32e3iT^{2} \)
79 \( 1 + 40iT - 6.24e3T^{2} \)
83 \( 1 + (-93 - 93i)T + 6.88e3iT^{2} \)
89 \( 1 - 30iT - 7.92e3T^{2} \)
97 \( 1 + (12 - 12i)T - 9.40e3iT^{2} \)
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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

−10.98816700885998946765037781109, −9.763698402816044382839187809700, −9.061516409740485338344520988386, −7.946680033049022121071910728037, −7.37623227060207952863675107331, −6.24793164507256641198846913758, −5.21717934895737897723979751904, −4.14065018367742846364065204204, −2.44953680416428328450775306204, −0.77792245781875828776542141108, 1.19029877160003910635911488839, 2.64197868092256633405935035009, 3.76251780683402980053370759450, 5.24063989221777173485652052904, 6.09959465441499361278566382897, 7.71580436606128620507935154608, 8.136274755116591802529441643843, 9.081880756409868057477351844910, 10.32077341497812238302166671179, 10.65070665597727047068646486991

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