Properties

Label 2-105-21.17-c3-0-30
Degree $2$
Conductor $105$
Sign $-0.818 + 0.573i$
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  + (3.53 − 2.04i)2-s + (−2.57 − 4.51i)3-s + (4.33 − 7.51i)4-s + (−2.5 − 4.33i)5-s + (−18.3 − 10.7i)6-s + (0.627 − 18.5i)7-s − 2.75i·8-s + (−13.7 + 23.2i)9-s + (−17.6 − 10.2i)10-s + (−7.33 − 4.23i)11-s + (−45.0 − 0.236i)12-s − 5.50i·13-s + (−35.5 − 66.7i)14-s + (−13.1 + 22.4i)15-s + (29.0 + 50.3i)16-s + (60.2 − 104. i)17-s + ⋯
L(s)  = 1  + (1.25 − 0.721i)2-s + (−0.495 − 0.868i)3-s + (0.542 − 0.938i)4-s + (−0.223 − 0.387i)5-s + (−1.24 − 0.728i)6-s + (0.0338 − 0.999i)7-s − 0.121i·8-s + (−0.509 + 0.860i)9-s + (−0.559 − 0.322i)10-s + (−0.200 − 0.116i)11-s + (−1.08 − 0.00568i)12-s − 0.117i·13-s + (−0.679 − 1.27i)14-s + (−0.225 + 0.386i)15-s + (0.454 + 0.786i)16-s + (0.860 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.818 + 0.573i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.818 + 0.573i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.692113 - 2.19379i\)
\(L(\frac12)\) \(\approx\) \(0.692113 - 2.19379i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.57 + 4.51i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-0.627 + 18.5i)T \)
good2 \( 1 + (-3.53 + 2.04i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (7.33 + 4.23i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.50iT - 2.19e3T^{2} \)
17 \( 1 + (-60.2 + 104. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-23.3 + 13.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (2.98 - 1.72i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 72.7iT - 2.43e4T^{2} \)
31 \( 1 + (-242. - 139. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-44.8 - 77.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 221.T + 6.89e4T^{2} \)
43 \( 1 - 495.T + 7.95e4T^{2} \)
47 \( 1 + (100. + 173. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (63.3 + 36.6i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-180. + 312. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (586. - 338. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (385. - 668. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 801. iT - 3.57e5T^{2} \)
73 \( 1 + (-881. - 509. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-96.8 - 167. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + (310. + 538. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 30.4iT - 9.12e5T^{2} \)
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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

−12.85806399340725586783120712354, −11.94810108095110708500304674751, −11.29507211275700081128750927692, −10.12977803831734715724591085258, −8.153236921108248750917064862299, −7.03470074201040815275767069620, −5.54244273035957823693046162552, −4.53662576809045237713001353593, −2.91477320823091963292292248179, −1.00074380248390908147649060169, 3.22146426263706670445063777858, 4.45214259641892685842251668420, 5.65131749810295110752002030500, 6.32778974532595850929509977801, 7.977654021766247819935131409481, 9.504161456575236801911759494221, 10.66760220658747276381851073636, 11.96608076721174184927536140390, 12.57223643679329540579240302851, 14.02662417340664973817446191249

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