Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.347 + 0.937i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.62 − 2.09i)7-s − 0.999i·8-s + (−2.59 + 1.5i)11-s − 2.44i·13-s + (−2.44 + 0.999i)14-s + (−0.5 + 0.866i)16-s + (0.507 + 0.878i)17-s + (−0.878 − 0.507i)19-s + 3·22-s + (3.67 + 2.12i)23-s + (−1.22 + 2.12i)26-s + (2.62 + 0.358i)28-s + 1.24i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.612 − 0.790i)7-s − 0.353i·8-s + (−0.783 + 0.452i)11-s − 0.679i·13-s + (−0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.123 + 0.213i)17-s + (−0.201 − 0.116i)19-s + 0.639·22-s + (0.766 + 0.442i)23-s + (−0.240 + 0.416i)26-s + (0.495 + 0.0677i)28-s + 0.230i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.347 + 0.937i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.347 + 0.937i)$
$L(1)$  $\approx$  $1.105779096$
$L(\frac12)$  $\approx$  $1.105779096$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.62 + 2.09i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (-0.507 - 0.878i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.12 + 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (-0.507 + 0.878i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.07 - 0.621i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.76 + 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (7.24 - 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 + 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−8.366516899476491193873984960053, −7.74017839819997984179307168047, −7.29751596520034587946887054675, −6.32554835365969881968027761586, −5.27169503413888698179441377210, −4.55496263285553851619221873351, −3.57809082300480419805992260702, −2.64053639490025127341516391349, −1.60402582523411515242624414103, −0.46521301317213635668169933435, 1.14657511021093436113004249526, 2.29022266079799523413965184335, 3.08422113787074866686839713766, 4.55067323334301733754103326046, 5.12037259095818188952337002888, 6.01626644500107315000586191836, 6.65345266803831546153183471778, 7.59642918319293815720609980552, 8.263116819859376050327486846572, 8.752742031851980234056398951953

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