Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.527 - 0.849i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 1.51i)2-s + (2.80 + 1.07i)3-s + (2.56 + 4.43i)4-s + (−1.93 − 1.11i)5-s + (5.70 + 7.04i)6-s + (−4.99 − 4.90i)7-s + 3.39i·8-s + (6.68 + 6.02i)9-s + (−3.37 − 5.84i)10-s + (−1.06 + 0.614i)11-s + (2.40 + 15.1i)12-s − 18.8·13-s + (−5.65 − 20.3i)14-s + (−4.22 − 5.21i)15-s + (5.11 − 8.86i)16-s + (16.5 − 9.57i)17-s + ⋯
L(s)  = 1  + (1.30 + 0.755i)2-s + (0.933 + 0.358i)3-s + (0.640 + 1.10i)4-s + (−0.387 − 0.223i)5-s + (0.950 + 1.17i)6-s + (−0.713 − 0.700i)7-s + 0.424i·8-s + (0.743 + 0.669i)9-s + (−0.337 − 0.584i)10-s + (−0.0967 + 0.0558i)11-s + (0.200 + 1.26i)12-s − 1.44·13-s + (−0.404 − 1.45i)14-s + (−0.281 − 0.347i)15-s + (0.319 − 0.554i)16-s + (0.975 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.527 - 0.849i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.527 - 0.849i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.52240 + 1.40249i\)
\(L(\frac12)\)  \(\approx\)  \(2.52240 + 1.40249i\)
\(L(2)\)   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 \( 1 + (-2.80 - 1.07i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (4.99 + 4.90i)T \)
good2 \( 1 + (-2.61 - 1.51i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (1.06 - 0.614i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 18.8T + 169T^{2} \)
17 \( 1 + (-16.5 + 9.57i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (10.0 - 17.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.4 - 9.46i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 31.9iT - 841T^{2} \)
31 \( 1 + (14.6 + 25.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.97 - 17.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 58.4iT - 1.68e3T^{2} \)
43 \( 1 - 57.9T + 1.84e3T^{2} \)
47 \( 1 + (41.1 + 23.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-3.20 + 1.85i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.3 - 12.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (41.8 - 72.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (21.3 + 37.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 0.779iT - 5.04e3T^{2} \)
73 \( 1 + (-4.35 - 7.54i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (31.2 - 54.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 75.5iT - 6.88e3T^{2} \)
89 \( 1 + (-73.5 - 42.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 133.T + 9.40e3T^{2} \)
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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

−13.93689789457439864287304921669, −12.90381995518640646102452646591, −12.24420069373667604464871144953, −10.34696412980211428966612533743, −9.372490541748080793635792777591, −7.68132831801174425374314781415, −7.08995424593175942014258586131, −5.32397411495714573225294135175, −4.17016227284908952121807275751, −3.12845352869365374780635566662, 2.42205576158440448782603963383, 3.30129702288492164518087663881, 4.74698282782565413565727877877, 6.33797950957403784360411971460, 7.73112408612040117852906234226, 9.130839132447738861847292996340, 10.32708175060459633520335274397, 11.75510158463901214577374248104, 12.60627037997234593818871531673, 13.06019749125358876429573322268

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