Properties

Degree 2
Conductor $ 3 \cdot 5 \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 + i·3-s + 4-s + (1 − 2i)5-s − 6-s + i·7-s + 3i·8-s − 9-s + (2 + i)10-s − 6·11-s + i·12-s − 2i·13-s − 14-s + (2 + i)15-s − 16-s − 4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (0.447 − 0.894i)5-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s − 1.80·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s + (0.516 + 0.258i)15-s − 0.250·16-s − 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (64, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.447 - 0.894i)$
$L(1)$  $\approx$  $0.987759 + 0.610468i$
$L(\frac12)$  $\approx$  $0.987759 + 0.610468i$
$L(\frac{3}{2})$   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 \( 1 - iT \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 - iT \)
good2 \( 1 - iT - 2T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−14.07765741036902913605419866632, −13.03878561228530477495552939210, −11.86948020238545760806007327417, −10.69192697832386041230196696032, −9.590614160303390257499736002489, −8.370119229283450604227174471651, −7.40285334522283855310331904915, −5.53092384774540902824306223888, −5.20934549967759721253207672094, −2.70173620439600678454391775595, 2.06135419027838012408223731157, 3.31926418578659151487141795523, 5.66673716415926716727854405042, 6.96295836787847011215570227142, 7.77449242194542902577294645997, 9.704083858777687059690286950373, 10.65843719583477371813672800041, 11.27433118061208772548131650751, 12.59576519015341727507158702442, 13.34777610896378407247861108327

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