Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.913 - 0.406i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (2.23 − 0.0466i)5-s + (−0.809 − 0.587i)6-s − 3.52i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (−2.11 + 0.735i)10-s + (1.62 + 4.99i)11-s + (0.951 + 0.309i)12-s + (−0.588 − 0.191i)13-s + (1.08 + 3.34i)14-s + (1.35 + 1.78i)15-s + (0.309 − 0.951i)16-s + (2.02 − 2.78i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (0.999 − 0.0208i)5-s + (−0.330 − 0.239i)6-s − 1.33i·7-s + (−0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + (−0.667 + 0.232i)10-s + (0.489 + 1.50i)11-s + (0.274 + 0.0892i)12-s + (−0.163 − 0.0530i)13-s + (0.290 + 0.895i)14-s + (0.349 + 0.459i)15-s + (0.0772 − 0.237i)16-s + (0.491 − 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.913 - 0.406i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.913 - 0.406i)$
$L(1)$  $\approx$  $1.03233 + 0.219321i$
$L(\frac12)$  $\approx$  $1.03233 + 0.219321i$
$L(\frac{3}{2})$   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 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (-2.23 + 0.0466i)T \)
good7 \( 1 + 3.52iT - 7T^{2} \)
11 \( 1 + (-1.62 - 4.99i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.588 + 0.191i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.02 + 2.78i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.83 - 1.33i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (8.51 - 2.76i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.16 - 1.57i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (7.90 + 5.74i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.952 + 0.309i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.94 + 5.98i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.51iT - 43T^{2} \)
47 \( 1 + (6.27 + 8.63i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.325 - 0.447i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0861 - 0.265i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.13 - 3.50i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.00 - 9.63i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.84 - 2.79i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.35 + 3.03i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.27 + 3.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.56 - 2.15i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.13 - 12.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.56 + 3.53i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.32407911235552733837067276265, −11.97471282169776881383486187279, −10.57201006734215199565741583667, −9.840287826060296677404417558749, −9.378899776135703111063011070545, −7.73096454491435614126478270379, −6.97578179596859319730062209694, −5.45280759704214018295464553695, −3.97036263420644048569315890827, −1.87744328217536809060412156640, 1.82178062140016157885121275801, 3.15059353067334317615240501357, 5.69283223682268175764259058830, 6.37042455923037834918133204637, 8.061709540707087754747429764543, 8.871151613178540654894506148576, 9.621429411489082631709238082761, 10.90001563583465098456301844079, 11.98034526068738761045359750917, 12.79874255017006211582642327824

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