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Properties

Label 2-450-15.2-c5-0-29
Degree $2$
Conductor $450$
Sign $-0.749 - 0.662i$
Analytic cond. $72.1727$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 − 2.82i)2-s − 16.0i·4-s + (−148. − 148. i)7-s + (−45.2 − 45.2i)8-s − 299. i·11-s + (193. − 193. i)13-s − 841.·14-s − 256.·16-s + (1.52e3 − 1.52e3i)17-s − 1.74e3i·19-s + (−847. − 847. i)22-s + (2.53e3 + 2.53e3i)23-s − 1.09e3i·26-s + (−2.37e3 + 2.37e3i)28-s − 2.61e3·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−1.14 − 1.14i)7-s + (−0.250 − 0.250i)8-s − 0.746i·11-s + (0.317 − 0.317i)13-s − 1.14·14-s − 0.250·16-s + (1.27 − 1.27i)17-s − 1.11i·19-s + (−0.373 − 0.373i)22-s + (1.00 + 1.00i)23-s − 0.317i·26-s + (−0.573 + 0.573i)28-s − 0.577·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.286557787\)
\(L(\frac12)\) \(\approx\) \(1.286557787\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 + 2.82i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (148. + 148. i)T + 1.68e4iT^{2} \)
11 \( 1 + 299. iT - 1.61e5T^{2} \)
13 \( 1 + (-193. + 193. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.52e3 + 1.52e3i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.74e3iT - 2.47e6T^{2} \)
23 \( 1 + (-2.53e3 - 2.53e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 2.61e3T + 2.05e7T^{2} \)
31 \( 1 + 8.65e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.37e3 - 5.37e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.58e3iT - 1.15e8T^{2} \)
43 \( 1 + (2.00e3 - 2.00e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.10e4 - 1.10e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (9.87e3 + 9.87e3i)T + 4.18e8iT^{2} \)
59 \( 1 + 1.39e4T + 7.14e8T^{2} \)
61 \( 1 + 2.81e4T + 8.44e8T^{2} \)
67 \( 1 + (2.41e4 + 2.41e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 6.51e3iT - 1.80e9T^{2} \)
73 \( 1 + (-6.15e4 + 6.15e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 4.24e4iT - 3.07e9T^{2} \)
83 \( 1 + (3.38e4 + 3.38e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 6.39e4T + 5.58e9T^{2} \)
97 \( 1 + (-5.89e4 - 5.89e4i)T + 8.58e9iT^{2} \)
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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

−9.703151673808824438883601649092, −9.272122358453156140418623097180, −7.66943767012848997215184442817, −6.91017313619803310912353543688, −5.83637696969325189242988454189, −4.82944124865422633253981573509, −3.41876659178962497914531174630, −3.10991152348246480736485719662, −1.15430848338211221797008039500, −0.27078184928953055793804800910, 1.80465609594179232276923259308, 3.11275100160679941495449328674, 4.00672823479679774812798922424, 5.46004289555157402972489242444, 6.03249898785349875658669665069, 6.97557588814955040990609684400, 8.058912334208620582530027862526, 9.012285074903897029954577480919, 9.782712922866023527173810129755, 10.82734732245766360554320175765

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