Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.447 - 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + i·7-s + i·8-s − 4·11-s + 2i·13-s + 14-s + 16-s − 2i·17-s + 4·19-s + 4i·22-s − 8i·23-s + 2·26-s i·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s − 1.20·11-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.485i·17-s + 0.917·19-s + 0.852i·22-s − 1.66i·23-s + 0.392·26-s − 0.188i·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.447 - 0.894i)$
$L(1)$  $\approx$  $0.8661965648$
$L(\frac12)$  $\approx$  $0.8661965648$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.906932641626885430617665078138, −8.186374640050119127973168262770, −7.45755297410796835159792071799, −6.53789868662020627382767062649, −5.57725204217112317878155124369, −4.89686538493475803700876431381, −4.13987840379394228599661793466, −2.88546078775332509877633844895, −2.49762955761122481548485427500, −1.14614896765759484954306572506, 0.29039028538893551807039851917, 1.75103076499376716873229677080, 3.14175913505525784010171094206, 3.78523889567437294335980661957, 5.09125125091688346379439405377, 5.35136070724991637364620316498, 6.26936768074248212635750847128, 7.34774345659382178971157875641, 7.56772470458726805704817209972, 8.416001800746905333477650856511

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