Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.997 + 0.0722i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (2.23 + 1.41i)7-s + 8-s − 1.41i·11-s + 5.39·13-s + (2.23 + 1.41i)14-s + 16-s − 2.23i·17-s − 1.30i·19-s − 1.41i·22-s + 23-s + 5.39·26-s + (2.23 + 1.41i)28-s − 9.24i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.845 + 0.534i)7-s + 0.353·8-s − 0.426i·11-s + 1.49·13-s + (0.597 + 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 0.300i·19-s − 0.301i·22-s + 0.208·23-s + 1.05·26-s + (0.422 + 0.267i)28-s − 1.71i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.997 + 0.0722i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.997 + 0.0722i)$
$L(1)$  $\approx$  $3.614578695$
$L(\frac12)$  $\approx$  $3.614578695$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 1.30iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 9.24iT - 29T^{2} \)
31 \( 1 - 8.56iT - 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + 4.08T + 41T^{2} \)
43 \( 1 + 6.41iT - 43T^{2} \)
47 \( 1 - 7.63iT - 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 5.39iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 - 5.01T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.573141849872687933813664921048, −7.996474276319849618718348697856, −7.05875476616026521591052489914, −6.22524189435961547558596296052, −5.59665449539923228691339021638, −4.86227924010914372572574346220, −4.00462302394114745083762719467, −3.14587158515161499169781840281, −2.16826248395092346880336251176, −1.07736682844406260445390677440, 1.19539301993949270472342004900, 2.00435793117497569964636886977, 3.37411254737377531120504563631, 3.96736676612888788265816003989, 4.80587964078635846851048088993, 5.54531945343789057849775741719, 6.41013100060320619263303871012, 7.04956028372084997325546588829, 8.011108427828764391562991249583, 8.419169690807411323745121858957

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