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Properties

Label 2-105-35.27-c3-0-1
Degree $2$
Conductor $105$
Sign $-0.416 - 0.909i$
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  + (−2.96 − 2.96i)2-s + (2.12 + 2.12i)3-s + 9.58i·4-s + (8.64 + 7.08i)5-s − 12.5i·6-s + (−18.0 − 4.10i)7-s + (4.70 − 4.70i)8-s + 8.99i·9-s + (−4.63 − 46.6i)10-s − 55.3·11-s + (−20.3 + 20.3i)12-s + (−43.2 − 43.2i)13-s + (41.3 + 65.7i)14-s + (3.31 + 33.3i)15-s + 48.7·16-s + (−75.6 + 75.6i)17-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)2-s + (0.408 + 0.408i)3-s + 1.19i·4-s + (0.773 + 0.633i)5-s − 0.856i·6-s + (−0.975 − 0.221i)7-s + (0.208 − 0.208i)8-s + 0.333i·9-s + (−0.146 − 1.47i)10-s − 1.51·11-s + (−0.489 + 0.489i)12-s + (−0.922 − 0.922i)13-s + (0.789 + 1.25i)14-s + (0.0570 + 0.574i)15-s + 0.762·16-s + (−1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.416 - 0.909i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.416 - 0.909i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.128635 + 0.200316i\)
\(L(\frac12)\) \(\approx\) \(0.128635 + 0.200316i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (-8.64 - 7.08i)T \)
7 \( 1 + (18.0 + 4.10i)T \)
good2 \( 1 + (2.96 + 2.96i)T + 8iT^{2} \)
11 \( 1 + 55.3T + 1.33e3T^{2} \)
13 \( 1 + (43.2 + 43.2i)T + 2.19e3iT^{2} \)
17 \( 1 + (75.6 - 75.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 23.6T + 6.85e3T^{2} \)
23 \( 1 + (-25.2 + 25.2i)T - 1.21e4iT^{2} \)
29 \( 1 - 280. iT - 2.43e4T^{2} \)
31 \( 1 + 120. iT - 2.97e4T^{2} \)
37 \( 1 + (267. + 267. i)T + 5.06e4iT^{2} \)
41 \( 1 - 185. iT - 6.89e4T^{2} \)
43 \( 1 + (133. - 133. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-280. + 280. i)T - 1.03e5iT^{2} \)
53 \( 1 + (99.3 - 99.3i)T - 1.48e5iT^{2} \)
59 \( 1 - 357.T + 2.05e5T^{2} \)
61 \( 1 - 228. iT - 2.26e5T^{2} \)
67 \( 1 + (-115. - 115. i)T + 3.00e5iT^{2} \)
71 \( 1 + 71.1T + 3.57e5T^{2} \)
73 \( 1 + (114. + 114. i)T + 3.89e5iT^{2} \)
79 \( 1 - 58.9iT - 4.93e5T^{2} \)
83 \( 1 + (-145. - 145. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + (-424. + 424. i)T - 9.12e5iT^{2} \)
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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

−13.23293111556290112008738230964, −12.63271091326065384878771376456, −10.77068755064208192876026620949, −10.43873265493556226244251185009, −9.658396057247774882817247599707, −8.586303727154812705629610326484, −7.25122903349787032979956893167, −5.51957176424789488765403578070, −3.20757665587196473475114488289, −2.32293816703141015277359536810, 0.16017680982419009292129421311, 2.47970399505853995627366513757, 5.17212727500801355206565888179, 6.48423142776940405657154460597, 7.37859373295274289893470122493, 8.612246679848143853484872610150, 9.428421868168465251943378832032, 10.10722931225691672810891062926, 12.11659346596384065619145079752, 13.19422223714525923038873029841

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