Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.139 + 0.990i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.90i·2-s + i·3-s − 1.62·4-s + (0.311 − 2.21i)5-s + 1.90·6-s i·7-s − 0.719i·8-s − 9-s + (−4.21 − 0.592i)10-s + 2·11-s − 1.62i·12-s + 6.42i·13-s − 1.90·14-s + (2.21 + 0.311i)15-s − 4.61·16-s + 4.42i·17-s + ⋯
L(s)  = 1  − 1.34i·2-s + 0.577i·3-s − 0.811·4-s + (0.139 − 0.990i)5-s + 0.776·6-s − 0.377i·7-s − 0.254i·8-s − 0.333·9-s + (−1.33 − 0.187i)10-s + 0.603·11-s − 0.468i·12-s + 1.78i·13-s − 0.508·14-s + (0.571 + 0.0803i)15-s − 1.15·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.139 + 0.990i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (64, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.139 + 0.990i)$
$L(1)$  $\approx$  $0.691672 - 0.795645i$
$L(\frac12)$  $\approx$  $0.691672 - 0.795645i$
$L(\frac{3}{2})$   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 \( 1 + (-0.311 + 2.21i)T \)
7 \( 1 + iT \)
good2 \( 1 + 1.90iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 6.42iT - 13T^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
23 \( 1 + 1.37iT - 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 - 5.18T + 31T^{2} \)
37 \( 1 - 7.61iT - 37T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 - 2.75iT - 47T^{2} \)
53 \( 1 + 9.18iT - 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 - 2.75iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 1.57iT - 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−13.30769290291366378232357486059, −12.08270890643338032845338257683, −11.56346165138034547950714484030, −10.29807747160142773847567369142, −9.467634483453780020428146994259, −8.585556909406263491608478922031, −6.57151319130391681096079613420, −4.66538171949640035486101255138, −3.76884378217202253573176078288, −1.65761090081185543965036771449, 2.89337844562107903556268616422, 5.35059424126018785705391587969, 6.29451319876752336862472580399, 7.30837504742868795446581709238, 8.084595345053140324737395304580, 9.504637956116825521260435185910, 10.95657576744041577766152525330, 12.03241079184143613288169175123, 13.45829095224240761971630202673, 14.23674687897973100047271979737

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