Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.949 + 0.313i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + (1 − 1.41i)3-s + 0.999·4-s + (−1.73 + 1.41i)5-s + (1.73 − 2.44i)6-s + (−1 + 2.44i)7-s − 1.73·8-s + (−1.00 − 2.82i)9-s + (−2.99 + 2.44i)10-s − 2.82i·11-s + (0.999 − 1.41i)12-s + 4·13-s + (−1.73 + 4.24i)14-s + (0.267 + 3.86i)15-s − 5·16-s + 2.82i·17-s + ⋯
L(s)  = 1  + 1.22·2-s + (0.577 − 0.816i)3-s + 0.499·4-s + (−0.774 + 0.632i)5-s + (0.707 − 0.999i)6-s + (−0.377 + 0.925i)7-s − 0.612·8-s + (−0.333 − 0.942i)9-s + (−0.948 + 0.774i)10-s − 0.852i·11-s + (0.288 − 0.408i)12-s + 1.10·13-s + (−0.462 + 1.13i)14-s + (0.0691 + 0.997i)15-s − 1.25·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.949 + 0.313i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (104, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.949 + 0.313i)$
$L(1)$  $\approx$  $1.69185 - 0.271605i$
$L(\frac12)$  $\approx$  $1.69185 - 0.271605i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
7 \( 1 + (1 - 2.44i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + 9.79iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 9.79iT - 61T^{2} \)
67 \( 1 + 4.89iT - 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−13.57243873297159201483064218059, −12.91383202839479147883208731616, −11.94039509481470754801583399537, −11.11844524753782811890678657124, −9.059678842927342555093971210870, −8.188854155736163205269556742399, −6.61545921427657028036280746754, −5.82147678121895778841274969685, −3.77425723124847594775547346659, −2.87845056192373894188141371571, 3.35833776691349154454458590662, 4.22850298330842497974047409690, 5.14419943791786367584315889687, 6.99425896532514950415498876883, 8.442539820864108846324136547569, 9.501650525825397968690254685471, 10.81670185047741558198465204750, 11.96268792922019792594974489279, 13.11647508764656751678784743138, 13.72557509293963817535519269605

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