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Properties

Label 2-105-21.17-c3-0-31
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: $\chi_{105} (101, \cdot )$
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

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

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