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Properties

Label 2-450-15.2-c3-0-14
Degree $2$
Conductor $450$
Sign $-0.374 + 0.927i$
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 + (4 + 4i)7-s + (5.65 + 5.65i)8-s − 45.2i·11-s + (21 − 21i)13-s − 11.3·14-s − 16.0·16-s + (−79.1 + 79.1i)17-s + 28i·19-s + (64.0 + 64.0i)22-s + (−65.0 − 65.0i)23-s + 59.3i·26-s + (16.0 − 16.0i)28-s − 111.·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.215 + 0.215i)7-s + (0.250 + 0.250i)8-s − 1.24i·11-s + (0.448 − 0.448i)13-s − 0.215·14-s − 0.250·16-s + (−1.12 + 1.12i)17-s + 0.338i·19-s + (0.620 + 0.620i)22-s + (−0.589 − 0.589i)23-s + 0.448i·26-s + (0.107 − 0.107i)28-s − 0.715·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ -0.374 + 0.927i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5537949544\)
\(L(\frac12)\) \(\approx\) \(0.5537949544\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4 - 4i)T + 343iT^{2} \)
11 \( 1 + 45.2iT - 1.33e3T^{2} \)
13 \( 1 + (-21 + 21i)T - 2.19e3iT^{2} \)
17 \( 1 + (79.1 - 79.1i)T - 4.91e3iT^{2} \)
19 \( 1 - 28iT - 6.85e3T^{2} \)
23 \( 1 + (65.0 + 65.0i)T + 1.21e4iT^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 - 304T + 2.97e4T^{2} \)
37 \( 1 + (231 + 231i)T + 5.06e4iT^{2} \)
41 \( 1 - 179. iT - 6.89e4T^{2} \)
43 \( 1 + (180 - 180i)T - 7.95e4iT^{2} \)
47 \( 1 + (127. - 127. i)T - 1.03e5iT^{2} \)
53 \( 1 + (359. + 359. i)T + 1.48e5iT^{2} \)
59 \( 1 + 752.T + 2.05e5T^{2} \)
61 \( 1 + 180T + 2.26e5T^{2} \)
67 \( 1 + (128 + 128i)T + 3.00e5iT^{2} \)
71 \( 1 + 1.02e3iT - 3.57e5T^{2} \)
73 \( 1 + (-683 + 683i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.39e3iT - 4.93e5T^{2} \)
83 \( 1 + (-186. - 186. i)T + 5.71e5iT^{2} \)
89 \( 1 + 448.T + 7.04e5T^{2} \)
97 \( 1 + (141 + 141i)T + 9.12e5iT^{2} \)
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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

−10.50109938611711301717496243776, −9.297734526507112925486559786861, −8.383924381766278673435463162800, −7.993708647362104988649203381643, −6.44182640856638164381635864714, −6.00438857968577251997542113205, −4.70859947674425548870819931346, −3.36552964748591787322843742190, −1.76657125273376891833697618895, −0.20953572585002551344740911443, 1.48307428848017155722015173553, 2.62710762538454609241569662355, 4.10065444703948398478001091343, 4.95873034299016993664407129882, 6.59190966762245362687747538513, 7.31607249073989269269596629979, 8.365716057511419495777704334542, 9.321664527307390416879981663692, 9.964720314198795905026221156580, 10.97956644081303604128305111106

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