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Properties

Label 2-450-15.8-c3-0-13
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 + (−21.9 + 21.9i)7-s + (−5.65 + 5.65i)8-s − 64.9i·11-s + (−13.0 − 13.0i)13-s − 61.9·14-s − 16.0·16-s + (5.53 + 5.53i)17-s − 103. i·19-s + (91.8 − 91.8i)22-s + (−19.7 + 19.7i)23-s − 37.0i·26-s + (−87.6 − 87.6i)28-s + 184.·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−1.18 + 1.18i)7-s + (−0.250 + 0.250i)8-s − 1.77i·11-s + (−0.279 − 0.279i)13-s − 1.18·14-s − 0.250·16-s + (0.0789 + 0.0789i)17-s − 1.25i·19-s + (0.889 − 0.889i)22-s + (−0.179 + 0.179i)23-s − 0.279i·26-s + (−0.591 − 0.591i)28-s + 1.18·29-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: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.081914272\)
\(L(\frac12)\) \(\approx\) \(1.081914272\)
\(L(\frac{5}{2})\) 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 \( 1 + (21.9 - 21.9i)T - 343iT^{2} \)
11 \( 1 + 64.9iT - 1.33e3T^{2} \)
13 \( 1 + (13.0 + 13.0i)T + 2.19e3iT^{2} \)
17 \( 1 + (-5.53 - 5.53i)T + 4.91e3iT^{2} \)
19 \( 1 + 103. iT - 6.85e3T^{2} \)
23 \( 1 + (19.7 - 19.7i)T - 1.21e4iT^{2} \)
29 \( 1 - 184.T + 2.43e4T^{2} \)
31 \( 1 - 19.8T + 2.97e4T^{2} \)
37 \( 1 + (-254. + 254. i)T - 5.06e4iT^{2} \)
41 \( 1 - 63.1iT - 6.89e4T^{2} \)
43 \( 1 + (104. + 104. i)T + 7.95e4iT^{2} \)
47 \( 1 + (185. + 185. i)T + 1.03e5iT^{2} \)
53 \( 1 + (259. - 259. i)T - 1.48e5iT^{2} \)
59 \( 1 + 255.T + 2.05e5T^{2} \)
61 \( 1 + 894.T + 2.26e5T^{2} \)
67 \( 1 + (-100. + 100. i)T - 3.00e5iT^{2} \)
71 \( 1 + 491. iT - 3.57e5T^{2} \)
73 \( 1 + (603. + 603. i)T + 3.89e5iT^{2} \)
79 \( 1 + 698. iT - 4.93e5T^{2} \)
83 \( 1 + (-707. + 707. i)T - 5.71e5iT^{2} \)
89 \( 1 - 389.T + 7.04e5T^{2} \)
97 \( 1 + (-29.2 + 29.2i)T - 9.12e5iT^{2} \)
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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

−10.58619960121510744400761967093, −9.276032954558248156588736108418, −8.779889233845063317884738320961, −7.71825873167017423458749588283, −6.34919968752000463356834857182, −6.00650308465063621667678498145, −4.90463559162822014657582298237, −3.34499012527427978434463542398, −2.71287389125027936107721113643, −0.29518845432573665081962820417, 1.39206146047096398880224134743, 2.84961737594806108945941764339, 4.03196416671537985239375287664, 4.73108859278038293391881241345, 6.27344436934636642905963892863, 6.95830869390499601239909318912, 7.945722573099279322061338338306, 9.703513661208391751287268114707, 9.837240482695568877403700965904, 10.69286529354926616008761722131

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