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Properties

Label 2-450-15.2-c3-0-11
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 + (12.1 + 12.1i)7-s + (−5.65 − 5.65i)8-s + 4.24i·11-s + (42.3 − 42.3i)13-s + 34.2·14-s − 16.0·16-s + (−17.4 + 17.4i)17-s + 36.4i·19-s + (6 + 6i)22-s + (44.0 + 44.0i)23-s − 119. i·26-s + (48.4 − 48.4i)28-s + 260.·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.654 + 0.654i)7-s + (−0.250 − 0.250i)8-s + 0.116i·11-s + (0.903 − 0.903i)13-s + 0.654·14-s − 0.250·16-s + (−0.249 + 0.249i)17-s + 0.440i·19-s + (0.0581 + 0.0581i)22-s + (0.398 + 0.398i)23-s − 0.903i·26-s + (0.327 − 0.327i)28-s + 1.67·29-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: $\chi_{450} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 0.733 + 0.679i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.879477493\)
\(L(\frac12)\) \(\approx\) \(2.879477493\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.1 - 12.1i)T + 343iT^{2} \)
11 \( 1 - 4.24iT - 1.33e3T^{2} \)
13 \( 1 + (-42.3 + 42.3i)T - 2.19e3iT^{2} \)
17 \( 1 + (17.4 - 17.4i)T - 4.91e3iT^{2} \)
19 \( 1 - 36.4iT - 6.85e3T^{2} \)
23 \( 1 + (-44.0 - 44.0i)T + 1.21e4iT^{2} \)
29 \( 1 - 260.T + 2.43e4T^{2} \)
31 \( 1 - 239.T + 2.97e4T^{2} \)
37 \( 1 + (-1.73 - 1.73i)T + 5.06e4iT^{2} \)
41 \( 1 + 252. iT - 6.89e4T^{2} \)
43 \( 1 + (-337. + 337. i)T - 7.95e4iT^{2} \)
47 \( 1 + (361. - 361. i)T - 1.03e5iT^{2} \)
53 \( 1 + (451. + 451. i)T + 1.48e5iT^{2} \)
59 \( 1 - 150.T + 2.05e5T^{2} \)
61 \( 1 - 859.T + 2.26e5T^{2} \)
67 \( 1 + (-136. - 136. i)T + 3.00e5iT^{2} \)
71 \( 1 + 464. iT - 3.57e5T^{2} \)
73 \( 1 + (-136. + 136. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.03e3iT - 4.93e5T^{2} \)
83 \( 1 + (755. + 755. i)T + 5.71e5iT^{2} \)
89 \( 1 - 432.T + 7.04e5T^{2} \)
97 \( 1 + (250. + 250. i)T + 9.12e5iT^{2} \)
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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

−10.70633853344084133339009082323, −9.898007277855565749750357995016, −8.674221606317106536669620625933, −8.059040264338168741136807540311, −6.59755030949861154318572155629, −5.64962047389760307511604650425, −4.77359145234473393925399713966, −3.54893368475272797535997977027, −2.37883317113806782123203720198, −1.04106582342113347158561084727, 1.12168942244280770386708469947, 2.82212807443790191983406480821, 4.23460492995547935170030729843, 4.81738547362619893560092332436, 6.25246466729614351647598562344, 6.88345548929537823162743434899, 8.036864416812788987738637677165, 8.700021874959332597798630176233, 9.881099898986266716180956107063, 11.04488888940688437390189698013

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