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Properties

Label 2-105-105.44-c2-0-9
Degree $2$
Conductor $105$
Sign $0.786 - 0.617i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.786 - 0.617i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56877 + 0.542779i\)
\(L(\frac12)\) \(\approx\) \(1.56877 + 0.542779i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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