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Properties

Label 2-105-21.20-c3-0-16
Degree $2$
Conductor $105$
Sign $0.362 + 0.932i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.948i·2-s + (−4.24 − 3.00i)3-s + 7.09·4-s + 5·5-s + (−2.84 + 4.02i)6-s + (15.4 + 10.2i)7-s − 14.3i·8-s + (8.96 + 25.4i)9-s − 4.74i·10-s − 22.6i·11-s + (−30.1 − 21.3i)12-s − 64.3i·13-s + (9.68 − 14.6i)14-s + (−21.2 − 15.0i)15-s + 43.2·16-s + 9.57·17-s + ⋯
L(s)  = 1  − 0.335i·2-s + (−0.816 − 0.577i)3-s + 0.887·4-s + 0.447·5-s + (−0.193 + 0.273i)6-s + (0.834 + 0.551i)7-s − 0.633i·8-s + (0.331 + 0.943i)9-s − 0.150i·10-s − 0.621i·11-s + (−0.724 − 0.512i)12-s − 1.37i·13-s + (0.184 − 0.279i)14-s + (−0.364 − 0.258i)15-s + 0.675·16-s + 0.136·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.362 + 0.932i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.362 + 0.932i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41695 - 0.969596i\)
\(L(\frac12)\) \(\approx\) \(1.41695 - 0.969596i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.24 + 3.00i)T \)
5 \( 1 - 5T \)
7 \( 1 + (-15.4 - 10.2i)T \)
good2 \( 1 + 0.948iT - 8T^{2} \)
11 \( 1 + 22.6iT - 1.33e3T^{2} \)
13 \( 1 + 64.3iT - 2.19e3T^{2} \)
17 \( 1 - 9.57T + 4.91e3T^{2} \)
19 \( 1 + 13.8iT - 6.85e3T^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 - 194. iT - 2.43e4T^{2} \)
31 \( 1 - 207. iT - 2.97e4T^{2} \)
37 \( 1 + 171.T + 5.06e4T^{2} \)
41 \( 1 - 214.T + 6.89e4T^{2} \)
43 \( 1 + 322.T + 7.95e4T^{2} \)
47 \( 1 - 582.T + 1.03e5T^{2} \)
53 \( 1 - 534. iT - 1.48e5T^{2} \)
59 \( 1 + 324.T + 2.05e5T^{2} \)
61 \( 1 + 32.4iT - 2.26e5T^{2} \)
67 \( 1 + 781.T + 3.00e5T^{2} \)
71 \( 1 + 357. iT - 3.57e5T^{2} \)
73 \( 1 - 925. iT - 3.89e5T^{2} \)
79 \( 1 - 827.T + 4.93e5T^{2} \)
83 \( 1 + 131.T + 5.71e5T^{2} \)
89 \( 1 + 505.T + 7.04e5T^{2} \)
97 \( 1 - 86.3iT - 9.12e5T^{2} \)
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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

−12.64025196186135780469354228415, −12.13167453695284174213791521654, −10.84897399030998406342451059685, −10.53232543386868785159679354188, −8.552634712707427367384737156191, −7.35735278006847762523789920014, −6.12052470480439061075631417244, −5.20551707657935520281363721432, −2.72480041717767630348499907797, −1.22101737539451337674045850702, 1.76607806372936291799274295215, 4.17970700069440692832989615694, 5.48624031453969302945980205851, 6.62991378937448790265327696418, 7.64601702218587024201069016830, 9.392922015207007426611423147683, 10.43066854428291136392284136087, 11.43475930884947445019500057192, 11.98569758330432500344367711800, 13.65874705030360215285868442885

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