Properties

Label 2-450-15.8-c3-0-12
Degree $2$
Conductor $450$
Sign $-0.733 + 0.679i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 1.41i)2-s + 4.00i·4-s + (0.123 − 0.123i)7-s + (5.65 − 5.65i)8-s − 4.24i·11-s + (−5.62 − 5.62i)13-s − 0.349·14-s − 16.0·16-s + (−34.4 − 34.4i)17-s + 110. i·19-s + (−6 + 6i)22-s + (111. − 111. i)23-s + 15.9i·26-s + (0.494 + 0.494i)28-s + 53.0·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.00668 − 0.00668i)7-s + (0.250 − 0.250i)8-s − 0.116i·11-s + (−0.120 − 0.120i)13-s − 0.00668·14-s − 0.250·16-s + (−0.491 − 0.491i)17-s + 1.33i·19-s + (−0.0581 + 0.0581i)22-s + (1.01 − 1.01i)23-s + 0.120i·26-s + (0.00334 + 0.00334i)28-s + 0.339·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -0.733 + 0.679i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8274904398\)
\(L(\frac12)\) \(\approx\) \(0.8274904398\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.123 + 0.123i)T - 343iT^{2} \)
11 \( 1 + 4.24iT - 1.33e3T^{2} \)
13 \( 1 + (5.62 + 5.62i)T + 2.19e3iT^{2} \)
17 \( 1 + (34.4 + 34.4i)T + 4.91e3iT^{2} \)
19 \( 1 - 110. iT - 6.85e3T^{2} \)
23 \( 1 + (-111. + 111. i)T - 1.21e4iT^{2} \)
29 \( 1 - 53.0T + 2.43e4T^{2} \)
31 \( 1 + 201.T + 2.97e4T^{2} \)
37 \( 1 + (-169. + 169. i)T - 5.06e4iT^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 + (202. + 202. i)T + 7.95e4iT^{2} \)
47 \( 1 + (106. + 106. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-243. + 243. i)T - 1.48e5iT^{2} \)
59 \( 1 + 57.2T + 2.05e5T^{2} \)
61 \( 1 + 609.T + 2.26e5T^{2} \)
67 \( 1 + (-340. + 340. i)T - 3.00e5iT^{2} \)
71 \( 1 + 990. iT - 3.57e5T^{2} \)
73 \( 1 + (847. + 847. i)T + 3.89e5iT^{2} \)
79 \( 1 - 436. iT - 4.93e5T^{2} \)
83 \( 1 + (127. - 127. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.23e3T + 7.04e5T^{2} \)
97 \( 1 + (202. - 202. i)T - 9.12e5iT^{2} \)
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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

−10.46045180468619238674856928361, −9.411285511017107964661425242091, −8.685277593989665365715729675994, −7.70436145652938695436161876761, −6.77734232364915417437945308982, −5.53807654292524077304964566872, −4.28717654892731515081905278750, −3.11286048505319745635359214437, −1.85410553494187980939267057968, −0.33140316256650815120004996656, 1.30958030922029572195782253826, 2.84749279088169841229343920601, 4.41723741701365729358879559460, 5.40911756423933895223247917029, 6.57353999929168357518471512564, 7.28552349013441410703750078025, 8.340583488249514958477700558276, 9.170493938134585980872808891496, 9.913310611597079043241186514148, 11.03504812511723836672086838058

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