Properties

Label 2-105-21.5-c3-0-8
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: $\chi_{105} (26, \cdot )$
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

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

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