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Properties

Label 2-105-35.27-c3-0-2
Degree $2$
Conductor $105$
Sign $-0.194 - 0.980i$
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  + (−0.817 − 0.817i)2-s + (2.12 + 2.12i)3-s − 6.66i·4-s + (−8.83 + 6.85i)5-s − 3.46i·6-s + (−4.27 + 18.0i)7-s + (−11.9 + 11.9i)8-s + 8.99i·9-s + (12.8 + 1.61i)10-s − 14.7·11-s + (14.1 − 14.1i)12-s + (44.4 + 44.4i)13-s + (18.2 − 11.2i)14-s + (−33.2 − 4.18i)15-s − 33.7·16-s + (−47.4 + 47.4i)17-s + ⋯
L(s)  = 1  + (−0.289 − 0.289i)2-s + (0.408 + 0.408i)3-s − 0.832i·4-s + (−0.789 + 0.613i)5-s − 0.235i·6-s + (−0.231 + 0.972i)7-s + (−0.529 + 0.529i)8-s + 0.333i·9-s + (0.405 + 0.0509i)10-s − 0.404·11-s + (0.340 − 0.340i)12-s + (0.947 + 0.947i)13-s + (0.347 − 0.214i)14-s + (−0.572 − 0.0720i)15-s − 0.526·16-s + (−0.677 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.194 - 0.980i$
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: \((2,\ 105,\ (\ :3/2),\ -0.194 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.547578 + 0.666569i\)
\(L(\frac12)\) \(\approx\) \(0.547578 + 0.666569i\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.83 - 6.85i)T \)
7 \( 1 + (4.27 - 18.0i)T \)
good2 \( 1 + (0.817 + 0.817i)T + 8iT^{2} \)
11 \( 1 + 14.7T + 1.33e3T^{2} \)
13 \( 1 + (-44.4 - 44.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (47.4 - 47.4i)T - 4.91e3iT^{2} \)
19 \( 1 - 9.66T + 6.85e3T^{2} \)
23 \( 1 + (59.3 - 59.3i)T - 1.21e4iT^{2} \)
29 \( 1 - 228. iT - 2.43e4T^{2} \)
31 \( 1 - 10.3iT - 2.97e4T^{2} \)
37 \( 1 + (198. + 198. i)T + 5.06e4iT^{2} \)
41 \( 1 + 345. iT - 6.89e4T^{2} \)
43 \( 1 + (-333. + 333. i)T - 7.95e4iT^{2} \)
47 \( 1 + (152. - 152. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-348. + 348. i)T - 1.48e5iT^{2} \)
59 \( 1 + 593.T + 2.05e5T^{2} \)
61 \( 1 - 328. iT - 2.26e5T^{2} \)
67 \( 1 + (-626. - 626. i)T + 3.00e5iT^{2} \)
71 \( 1 - 360.T + 3.57e5T^{2} \)
73 \( 1 + (-504. - 504. i)T + 3.89e5iT^{2} \)
79 \( 1 - 159. iT - 4.93e5T^{2} \)
83 \( 1 + (-413. - 413. i)T + 5.71e5iT^{2} \)
89 \( 1 - 707.T + 7.04e5T^{2} \)
97 \( 1 + (604. - 604. i)T - 9.12e5iT^{2} \)
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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

−13.85272220353937279611109063630, −12.29727529844601968384519424962, −11.17921181508961746468400459616, −10.53139986126087522649983746730, −9.191860794735366817658943413984, −8.498769860960342905327748544018, −6.79613993848027710372959038072, −5.50115840417048171665195074368, −3.78984989426667239787739189532, −2.17367967256008508200383780439, 0.49513813189215361971275683486, 3.18726741129662998616517528462, 4.37250867908212691485337764111, 6.52254175114467619637501724640, 7.79397151612189176190971638577, 8.167235283190109195948559370855, 9.447622074216468755104777565864, 10.98056743150975621485855514927, 12.12699419713665212419330976660, 13.10457904004886611349304470608

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