Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.970 - 0.241i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.70i·2-s + i·3-s − 5.34·4-s + (2.17 + 0.539i)5-s − 2.70·6-s i·7-s − 9.04i·8-s − 9-s + (−1.46 + 5.87i)10-s + 2·11-s − 5.34i·12-s + 0.921i·13-s + 2.70·14-s + (−0.539 + 2.17i)15-s + 13.8·16-s − 1.07i·17-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.577i·3-s − 2.67·4-s + (0.970 + 0.241i)5-s − 1.10·6-s − 0.377i·7-s − 3.19i·8-s − 0.333·9-s + (−0.461 + 1.85i)10-s + 0.603·11-s − 1.54i·12-s + 0.255i·13-s + 0.724·14-s + (−0.139 + 0.560i)15-s + 3.45·16-s − 0.261i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.970 - 0.241i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (64, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.970 - 0.241i)$
$L(1)$  $\approx$  $0.120404 + 0.983921i$
$L(\frac12)$  $\approx$  $0.120404 + 0.983921i$
$L(\frac{3}{2})$   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 \( 1 + (-2.17 - 0.539i)T \)
7 \( 1 + iT \)
good2 \( 1 - 2.70iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.921iT - 13T^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 - 2.34iT - 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + 6.52iT - 43T^{2} \)
47 \( 1 + 4.68iT - 47T^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 + 4.68iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 + 8.34T + 89T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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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

−14.30010746691742266607213787971, −13.90176400871089569727227151612, −12.70220566872127802201624935902, −10.65276762931066666301626823265, −9.484670246199797267850012101935, −8.844067668680403265045968283277, −7.35766252171978515182272696079, −6.35057381328158010672196517790, −5.35710948013680642658277880643, −4.05511280869542593063179756039, 1.51994669840817750292699618157, 2.84094034813626633848091687740, 4.65243554777495446773253593720, 6.11461818596453356629325934372, 8.398271019182241148811939884979, 9.253200390506736368513878183527, 10.25221378914337473604070958923, 11.25244514403042399458738670214, 12.38084342986702324751065534177, 12.88196609135143912413296792723

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