Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.0226 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.358 + 2.62i)7-s − 0.999·8-s + (2.59 − 1.5i)11-s − 2.44·13-s + (2.44 + i)14-s + (−0.5 + 0.866i)16-s + (−5.12 + 2.95i)17-s + (5.12 + 2.95i)19-s − 3i·22-s + (−2.12 + 3.67i)23-s + (−1.22 + 2.12i)26-s + (2.09 − 1.62i)28-s − 7.24i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.135 + 0.990i)7-s − 0.353·8-s + (0.783 − 0.452i)11-s − 0.679·13-s + (0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (−1.24 + 0.717i)17-s + (1.17 + 0.678i)19-s − 0.639i·22-s + (−0.442 + 0.766i)23-s + (−0.240 + 0.416i)26-s + (0.395 − 0.306i)28-s − 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0226 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0226 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0226 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.0226 - 0.999i)$
$L(1)$  $\approx$  $1.055366109$
$L(\frac12)$  $\approx$  $1.055366109$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.358 - 2.62i)T \)
good11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + (5.12 - 2.95i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.12 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.210 - 0.121i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 0.242iT - 43T^{2} \)
47 \( 1 + (-5.12 - 2.95i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.62 + 6.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.03 - 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-0.717 - 1.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.63iT - 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−8.966351217043290632469896279040, −8.328724309629476444671506820256, −7.36569985433005758617454222443, −6.39881206279445751382378398583, −5.71305629366974180733138956867, −5.06062344840377603643035280165, −4.03887067095421861340744862363, −3.32578114745567266174791531967, −2.26654682358576379552450942281, −1.49258117889385334687748145519, 0.27476947326387253814728204320, 1.78126119592431311645194250637, 3.03290377059287517630615265191, 3.97725085879656491914827686551, 4.68780754015891569302459729848, 5.27649518454375107668419863488, 6.46015909779363909641759388153, 7.15685660649319293103865103187, 7.31675388562747713569791911121, 8.443021305902310283060558819102

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