Properties

Label 2-105-21.5-c3-0-15
Degree $2$
Conductor $105$
Sign $-0.125 - 0.992i$
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  + (3.95 + 2.28i)2-s + (3.88 + 3.45i)3-s + (6.41 + 11.1i)4-s + (−2.5 + 4.33i)5-s + (7.46 + 22.5i)6-s + (−15.3 − 10.3i)7-s + 22.0i·8-s + (3.14 + 26.8i)9-s + (−19.7 + 11.4i)10-s + (38.8 − 22.4i)11-s + (−13.4 + 65.2i)12-s − 21.9i·13-s + (−37.1 − 75.9i)14-s + (−24.6 + 8.17i)15-s + (1.01 − 1.75i)16-s + (−18.3 − 31.7i)17-s + ⋯
L(s)  = 1  + (1.39 + 0.806i)2-s + (0.747 + 0.664i)3-s + (0.801 + 1.38i)4-s + (−0.223 + 0.387i)5-s + (0.507 + 1.53i)6-s + (−0.830 − 0.557i)7-s + 0.974i·8-s + (0.116 + 0.993i)9-s + (−0.624 + 0.360i)10-s + (1.06 − 0.614i)11-s + (−0.323 + 1.57i)12-s − 0.468i·13-s + (−0.709 − 1.44i)14-s + (−0.424 + 0.140i)15-s + (0.0157 − 0.0273i)16-s + (−0.261 − 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.125 - 0.992i$
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: \((2,\ 105,\ (\ :3/2),\ -0.125 - 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.45037 + 2.77878i\)
\(L(\frac12)\) \(\approx\) \(2.45037 + 2.77878i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.88 - 3.45i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (15.3 + 10.3i)T \)
good2 \( 1 + (-3.95 - 2.28i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-38.8 + 22.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 21.9iT - 2.19e3T^{2} \)
17 \( 1 + (18.3 + 31.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-91.7 - 52.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (21.5 + 12.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 31.6iT - 2.43e4T^{2} \)
31 \( 1 + (262. - 151. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-130. + 226. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 294.T + 6.89e4T^{2} \)
43 \( 1 + 302.T + 7.95e4T^{2} \)
47 \( 1 + (-59.6 + 103. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-560. + 323. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-3.12 - 5.40i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (702. + 405. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-13.4 - 23.2i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 639. iT - 3.57e5T^{2} \)
73 \( 1 + (619. - 357. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (312. - 541. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 630.T + 5.71e5T^{2} \)
89 \( 1 + (350. - 607. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 528. iT - 9.12e5T^{2} \)
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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

−13.89883922192416527666329595756, −13.01213900198558329440325269454, −11.75165845502857782598226983001, −10.37216469295637685202982564818, −9.171921852687759910753699500096, −7.65636238484230689216737282131, −6.68906086712443946188083123756, −5.34661627295263801905763648891, −3.82667512475031859590790948718, −3.28085088675682315466624940502, 1.72047128427026300131665556761, 3.14717261271297024507648560313, 4.26011158595159588464696904892, 5.92018769416692050012463838610, 7.08175334245697302305335613463, 8.819924104482389292088956508845, 9.761446050705167078190448452774, 11.64144484024454781186422282148, 12.09543960792343447100688003714, 13.07444279324851038089536708057

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