Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.786 + 0.617i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.949 + 1.64i)2-s + (−0.107 − 2.99i)3-s + (0.196 − 0.340i)4-s + (−4.51 − 2.14i)5-s + (4.82 − 3.02i)6-s + (4.60 − 5.26i)7-s + 8.34·8-s + (−8.97 + 0.647i)9-s + (−0.759 − 9.46i)10-s + (8.35 + 4.82i)11-s + (−1.04 − 0.552i)12-s − 16.9i·13-s + (13.0 + 2.57i)14-s + (−5.94 + 13.7i)15-s + (7.13 + 12.3i)16-s + (−12.1 + 21.1i)17-s + ⋯
L(s)  = 1  + (0.474 + 0.822i)2-s + (−0.0359 − 0.999i)3-s + (0.0490 − 0.0850i)4-s + (−0.903 − 0.429i)5-s + (0.804 − 0.504i)6-s + (0.658 − 0.752i)7-s + 1.04·8-s + (−0.997 + 0.0718i)9-s + (−0.0759 − 0.946i)10-s + (0.759 + 0.438i)11-s + (−0.0867 − 0.0460i)12-s − 1.30i·13-s + (0.931 + 0.184i)14-s + (−0.396 + 0.918i)15-s + (0.446 + 0.772i)16-s + (−0.716 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.786 + 0.617i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.786 + 0.617i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.56877 - 0.542779i\)
\(L(\frac12)\)  \(\approx\)  \(1.56877 - 0.542779i\)
\(L(2)\)   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 \( 1 + (0.107 + 2.99i)T \)
5 \( 1 + (4.51 + 2.14i)T \)
7 \( 1 + (-4.60 + 5.26i)T \)
good2 \( 1 + (-0.949 - 1.64i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-8.35 - 4.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + (12.1 - 21.1i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.95 - 12.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.354 + 0.614i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 16.5iT - 841T^{2} \)
31 \( 1 + (-7.12 + 12.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.08 + 0.626i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 24.1iT - 1.68e3T^{2} \)
43 \( 1 - 57.6iT - 1.84e3T^{2} \)
47 \( 1 + (16.0 + 27.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-8.67 + 15.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-75.8 - 43.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-52.2 - 90.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (86.1 + 49.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 50.7iT - 5.04e3T^{2} \)
73 \( 1 + (-81.5 - 47.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.71 + 6.43i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 69.6T + 6.88e3T^{2} \)
89 \( 1 + (78.3 - 45.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 90.4iT - 9.40e3T^{2} \)
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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

−13.44340879099234689947378061904, −12.63769790478790380876440335082, −11.48189650832999500184723810680, −10.48454814566668209878448624751, −8.381594408998829818011661983867, −7.68205102389288101843672397541, −6.75178399161148811633966862534, −5.44801101701232487503991993257, −4.06568049767195798727193708652, −1.29721520850707136942936865128, 2.64237951544379438462387520572, 3.96556290800960314394350933186, 4.87691313155132419515159087491, 6.87237400702786226160381104051, 8.400317906475146877703434114639, 9.440256708321372866551545732848, 11.02820887421269709369340412322, 11.52731938855647127583756273343, 11.98287007650507002464016836248, 13.78848559773336610630185086291

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