Properties

Label 2-450-15.2-c3-0-17
Degree $2$
Conductor $450$
Sign $-0.927 - 0.374i$
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 + (−9 − 9i)7-s + (−5.65 − 5.65i)8-s − 38.1i·11-s + (9 − 9i)13-s − 25.4·14-s − 16.0·16-s + (−55.1 + 55.1i)17-s + 38i·19-s + (−54.0 − 54.0i)22-s + (29.6 + 29.6i)23-s − 25.4i·26-s + (−36.0 + 36.0i)28-s − 76.3·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.485 − 0.485i)7-s + (−0.250 − 0.250i)8-s − 1.04i·11-s + (0.192 − 0.192i)13-s − 0.485·14-s − 0.250·16-s + (−0.786 + 0.786i)17-s + 0.458i·19-s + (−0.523 − 0.523i)22-s + (0.269 + 0.269i)23-s − 0.192i·26-s + (−0.242 + 0.242i)28-s − 0.489·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(0.7853099853\)
\(L(\frac12)\) \(\approx\) \(0.7853099853\)
\(L(\frac{5}{2})\) 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 \( 1 + (9 + 9i)T + 343iT^{2} \)
11 \( 1 + 38.1iT - 1.33e3T^{2} \)
13 \( 1 + (-9 + 9i)T - 2.19e3iT^{2} \)
17 \( 1 + (55.1 - 55.1i)T - 4.91e3iT^{2} \)
19 \( 1 - 38iT - 6.85e3T^{2} \)
23 \( 1 + (-29.6 - 29.6i)T + 1.21e4iT^{2} \)
29 \( 1 + 76.3T + 2.43e4T^{2} \)
31 \( 1 + 236T + 2.97e4T^{2} \)
37 \( 1 + (279 + 279i)T + 5.06e4iT^{2} \)
41 \( 1 + 343. iT - 6.89e4T^{2} \)
43 \( 1 + (360 - 360i)T - 7.95e4iT^{2} \)
47 \( 1 + (254. - 254. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-352. - 352. i)T + 1.48e5iT^{2} \)
59 \( 1 - 343.T + 2.05e5T^{2} \)
61 \( 1 - 110T + 2.26e5T^{2} \)
67 \( 1 + (702 + 702i)T + 3.00e5iT^{2} \)
71 \( 1 + 76.3iT - 3.57e5T^{2} \)
73 \( 1 + (378 - 378i)T - 3.89e5iT^{2} \)
79 \( 1 + 272iT - 4.93e5T^{2} \)
83 \( 1 + (59.3 + 59.3i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + (684 + 684i)T + 9.12e5iT^{2} \)
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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

−10.52763136747876610523983156541, −9.344173272203479787107494526656, −8.510236042276687160167996973890, −7.26220973281247833834027090538, −6.21509301682748644268187553463, −5.38326080646385745449066775915, −3.97092990057339165679112267826, −3.28813772196762126830788766029, −1.74462721346389598649262197041, −0.19846036180063258910948768049, 2.06963226212020271946855033376, 3.32873846847170543318360745956, 4.59633212397918176774455916260, 5.41611405860576637532098272494, 6.70057991276880334571534958722, 7.14267238588512897097662773666, 8.489219021824306659191291016564, 9.251171309144406393787004462798, 10.18536453893052465522018873052, 11.43169332868831801302788431092

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