Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} $
Sign $0.886 + 0.461i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−1.22 − 1.22i)3-s + (0.866 + 0.499i)4-s + (0.866 + 1.49i)6-s + (−0.328 + 1.22i)7-s + (−0.707 − 0.707i)8-s + 2.99i·9-s + (3 − 1.73i)11-s + (−0.448 − 1.67i)12-s + (0.328 + 1.22i)13-s + (0.633 − 1.09i)14-s + (0.500 + 0.866i)16-s + (0.776 − 2.89i)18-s + 7.19i·19-s + (1.90 − 1.09i)21-s + (−3.34 + 0.896i)22-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.707 − 0.707i)3-s + (0.433 + 0.249i)4-s + (0.353 + 0.612i)6-s + (−0.124 + 0.462i)7-s + (−0.249 − 0.249i)8-s + 0.999i·9-s + (0.904 − 0.522i)11-s + (−0.129 − 0.482i)12-s + (0.0910 + 0.339i)13-s + (0.169 − 0.293i)14-s + (0.125 + 0.216i)16-s + (0.183 − 0.683i)18-s + 1.65i·19-s + (0.415 − 0.239i)21-s + (−0.713 + 0.191i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.886 + 0.461i$
motivic weight  =  \(1\)
character  :  $\chi_{450} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 450,\ (\ :1/2),\ 0.886 + 0.461i)\)
\(L(1)\)  \(\approx\)  \(0.808700 - 0.197992i\)
\(L(\frac12)\)  \(\approx\)  \(0.808700 - 0.197992i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (0.328 - 1.22i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.328 - 1.22i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 7.19iT - 19T^{2} \)
23 \( 1 + (-7.91 + 2.12i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.63 + 6.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.09 + 8.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.24 - 1.67i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.79 - 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.55 - 1.55i)T + 53iT^{2} \)
59 \( 1 + (-6.23 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.0 - 3.22i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.76 - 6.57i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (-3.79 + 14.1i)T + (-84.0 - 48.5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.21924615335350346397262789344, −10.16851754033603220895731867312, −9.203899083931438536654292800966, −8.285055013817634440684232513436, −7.39263103819416734665686420875, −6.33114234354064830004121734766, −5.73651428018237007185778287154, −4.10034308208664472557946152105, −2.45549830396006036687089945719, −1.05742311178366563993550327673, 0.996222834343404558890805546803, 3.16302568869081221084260960040, 4.50766125782078015708643201299, 5.45182631592828314617141548128, 6.79182171906068777760544147548, 7.15737692338919750702820663714, 8.920520361343068263845685345034, 9.205959087325125764909491973023, 10.38190300253415894713897037831, 10.89986066399891145259560615176

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