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 + (1.22 − 4.57i)7-s + (−0.707 − 0.707i)8-s + 2.99i·9-s + (3 − 1.73i)11-s + (0.448 + 1.67i)12-s + (−1.22 − 4.57i)13-s + (−2.36 + 4.09i)14-s + (0.500 + 0.866i)16-s + (0.776 − 2.89i)18-s − 3.19i·19-s + (7.09 − 4.09i)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 + (0.462 − 1.72i)7-s + (−0.249 − 0.249i)8-s + 0.999i·9-s + (0.904 − 0.522i)11-s + (0.129 + 0.482i)12-s + (−0.339 − 1.26i)13-s + (−0.632 + 1.09i)14-s + (0.125 + 0.216i)16-s + (0.183 − 0.683i)18-s − 0.733i·19-s + (1.54 − 0.894i)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  =  \((2,\ 450,\ (\ :1/2),\ 0.886 + 0.461i)\)
\(L(1)\)  \(\approx\)  \(1.32661 - 0.324792i\)
\(L(\frac12)\)  \(\approx\)  \(1.32661 - 0.324792i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.22 + 4.57i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 4.57i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + (2.12 - 0.568i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.36 - 9.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0980 - 0.169i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.448 - 0.120i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.79 - 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.79 + 5.79i)T + 53iT^{2} \)
59 \( 1 + (2.76 - 4.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.34 + 1.43i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + (-8.66 + 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.45 - 16.6i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (-0.688 + 2.56i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.66702125207365385270671051669, −10.24698770054042260667926710033, −9.286346332189886810226004788709, −8.319607046894964549788182581022, −7.65145542356140913712208226309, −6.70478880134960809713769029850, −5.00655968302684992990315883904, −3.93826200667636714244217401464, −2.98827104967234392344874016957, −1.11328187380056148250192053040, 1.75092288025863658103510584705, 2.49843011870983795322708962396, 4.27263304044983162287177812681, 5.89807137174929348587409681904, 6.58109383933437927308925966046, 7.72051686749188825945507746885, 8.488248452178607480773618381695, 9.244022547657620350280246816698, 9.747749674057188352070701436691, 11.46853783225992598242280996268

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