Properties

Label 2-105-21.5-c3-0-18
Degree $2$
Conductor $105$
Sign $0.913 + 0.406i$
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  + (0.155 + 0.0897i)2-s + (5.17 + 0.514i)3-s + (−3.98 − 6.90i)4-s + (−2.5 + 4.33i)5-s + (0.757 + 0.544i)6-s + (18.1 + 3.54i)7-s − 2.86i·8-s + (26.4 + 5.31i)9-s + (−0.777 + 0.448i)10-s + (42.8 − 24.7i)11-s + (−17.0 − 37.7i)12-s − 62.5i·13-s + (2.50 + 2.18i)14-s + (−15.1 + 21.1i)15-s + (−31.6 + 54.7i)16-s + (19.4 + 33.6i)17-s + ⋯
L(s)  = 1  + (0.0549 + 0.0317i)2-s + (0.995 + 0.0989i)3-s + (−0.497 − 0.862i)4-s + (−0.223 + 0.387i)5-s + (0.0515 + 0.0370i)6-s + (0.981 + 0.191i)7-s − 0.126i·8-s + (0.980 + 0.196i)9-s + (−0.0245 + 0.0141i)10-s + (1.17 − 0.677i)11-s + (−0.410 − 0.907i)12-s − 1.33i·13-s + (0.0478 + 0.0416i)14-s + (−0.260 + 0.363i)15-s + (−0.493 + 0.855i)16-s + (0.277 + 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.913 + 0.406i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.13710 - 0.453903i\)
\(L(\frac12)\) \(\approx\) \(2.13710 - 0.453903i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.17 - 0.514i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-18.1 - 3.54i)T \)
good2 \( 1 + (-0.155 - 0.0897i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-42.8 + 24.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 62.5iT - 2.19e3T^{2} \)
17 \( 1 + (-19.4 - 33.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-14.1 - 8.18i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (125. + 72.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 246. iT - 2.43e4T^{2} \)
31 \( 1 + (-128. + 74.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 429.T + 6.89e4T^{2} \)
43 \( 1 - 73.5T + 7.95e4T^{2} \)
47 \( 1 + (124. - 214. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (263. - 152. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (336. + 582. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (279. + 161. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (103. + 179. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 717. iT - 3.57e5T^{2} \)
73 \( 1 + (-84.3 + 48.7i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-450. + 779. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 90.9T + 5.71e5T^{2} \)
89 \( 1 + (-328. + 568. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.46e3iT - 9.12e5T^{2} \)
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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

−13.64220278024304702690311801906, −12.24617697289822251715660654398, −10.83613681842750276503973560278, −10.00503191496080437292090591325, −8.719435740445633512940736171406, −8.016331337146138028502222072348, −6.33882780639524779732745003835, −4.85620770377214408652164117466, −3.45026258385807959322490083848, −1.44095739361936607102003000227, 1.83445272968235678017430212006, 3.85166299193527752347953722473, 4.57562426235185469323027701335, 7.03468807619538435756618577898, 7.959603426402136027575634801169, 8.905500167193157656895184000582, 9.704708341738929064033560985662, 11.76248851961166384065124295438, 12.14127417869027157165849081982, 13.76450572304564095610160845599

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