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Properties

Label 2-450-15.2-c5-0-19
Degree $2$
Conductor $450$
Sign $0.279 + 0.960i$
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 + (140. + 140. i)7-s + (−45.2 − 45.2i)8-s − 540. i·11-s + (−798. + 798. i)13-s + 795.·14-s − 256.·16-s + (598. − 598. i)17-s − 1.78e3i·19-s + (−1.53e3 − 1.53e3i)22-s + (−718. − 718. i)23-s + 4.51e3i·26-s + (2.25e3 − 2.25e3i)28-s + 5.38e3·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (1.08 + 1.08i)7-s + (−0.250 − 0.250i)8-s − 1.34i·11-s + (−1.31 + 1.31i)13-s + 1.08·14-s − 0.250·16-s + (0.502 − 0.502i)17-s − 1.13i·19-s + (−0.673 − 0.673i)22-s + (−0.283 − 0.283i)23-s + 1.31i·26-s + (0.542 − 0.542i)28-s + 1.18·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.279 + 0.960i$
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: \((2,\ 450,\ (\ :5/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.978843664\)
\(L(\frac12)\) \(\approx\) \(2.978843664\)
\(L(\frac{7}{2})\) 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 \( 1 + (-140. - 140. i)T + 1.68e4iT^{2} \)
11 \( 1 + 540. iT - 1.61e5T^{2} \)
13 \( 1 + (798. - 798. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-598. + 598. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.78e3iT - 2.47e6T^{2} \)
23 \( 1 + (718. + 718. i)T + 6.43e6iT^{2} \)
29 \( 1 - 5.38e3T + 2.05e7T^{2} \)
31 \( 1 - 1.80e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.34e3 - 7.34e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.24e4iT - 1.15e8T^{2} \)
43 \( 1 + (-9.90e3 + 9.90e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.82e3 - 1.82e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.60e4 - 1.60e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 4.52e4T + 7.14e8T^{2} \)
61 \( 1 - 605.T + 8.44e8T^{2} \)
67 \( 1 + (2.30e4 + 2.30e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 1.13e4iT - 1.80e9T^{2} \)
73 \( 1 + (-4.18e4 + 4.18e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 6.19e4iT - 3.07e9T^{2} \)
83 \( 1 + (8.21e4 + 8.21e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.07e4T + 5.58e9T^{2} \)
97 \( 1 + (-5.14e4 - 5.14e4i)T + 8.58e9iT^{2} \)
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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

−10.23769713697687287842946998464, −9.117829523111680507545531186939, −8.547507234787600162848417730263, −7.28789317898799639560138205349, −6.10741519910525799174481970413, −5.13378444802171753806459211133, −4.44519349930527947629297341986, −2.86772775921176826106490365456, −2.12367778342954439753567852657, −0.68078773405721461219423784954, 1.05915917745943889675923354193, 2.44568659456117531409909804209, 3.94841014716598271711997661918, 4.72513075653834674034156283268, 5.57519218329894079219666062632, 6.93342257638079032678670017333, 7.77966497262555624168915532975, 8.085596377217310633041756428831, 9.906444971471011890883965881652, 10.24753496985897237353138481664

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