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Properties

Label 2-105-35.13-c3-0-20
Degree $2$
Conductor $105$
Sign $-0.448 + 0.894i$
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  + (2.53 − 2.53i)2-s + (2.12 − 2.12i)3-s − 4.87i·4-s + (2.35 − 10.9i)5-s − 10.7i·6-s + (−16.3 − 8.62i)7-s + (7.91 + 7.91i)8-s − 8.99i·9-s + (−21.7 − 33.7i)10-s + 10.7·11-s + (−10.3 − 10.3i)12-s + (3.91 − 3.91i)13-s + (−63.4 + 19.7i)14-s + (−18.1 − 28.1i)15-s + 79.2·16-s + (10.8 + 10.8i)17-s + ⋯
L(s)  = 1  + (0.897 − 0.897i)2-s + (0.408 − 0.408i)3-s − 0.609i·4-s + (0.210 − 0.977i)5-s − 0.732i·6-s + (−0.884 − 0.465i)7-s + (0.349 + 0.349i)8-s − 0.333i·9-s + (−0.688 − 1.06i)10-s + 0.293·11-s + (−0.249 − 0.249i)12-s + (0.0835 − 0.0835i)13-s + (−1.21 + 0.376i)14-s + (−0.313 − 0.485i)15-s + 1.23·16-s + (0.154 + 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.448 + 0.894i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.448 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43365 - 2.32212i\)
\(L(\frac12)\) \(\approx\) \(1.43365 - 2.32212i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.35 + 10.9i)T \)
7 \( 1 + (16.3 + 8.62i)T \)
good2 \( 1 + (-2.53 + 2.53i)T - 8iT^{2} \)
11 \( 1 - 10.7T + 1.33e3T^{2} \)
13 \( 1 + (-3.91 + 3.91i)T - 2.19e3iT^{2} \)
17 \( 1 + (-10.8 - 10.8i)T + 4.91e3iT^{2} \)
19 \( 1 - 27.5T + 6.85e3T^{2} \)
23 \( 1 + (-38.4 - 38.4i)T + 1.21e4iT^{2} \)
29 \( 1 - 179. iT - 2.43e4T^{2} \)
31 \( 1 - 228. iT - 2.97e4T^{2} \)
37 \( 1 + (-287. + 287. i)T - 5.06e4iT^{2} \)
41 \( 1 + 63.4iT - 6.89e4T^{2} \)
43 \( 1 + (-95.7 - 95.7i)T + 7.95e4iT^{2} \)
47 \( 1 + (-144. - 144. i)T + 1.03e5iT^{2} \)
53 \( 1 + (173. + 173. i)T + 1.48e5iT^{2} \)
59 \( 1 + 194.T + 2.05e5T^{2} \)
61 \( 1 - 310. iT - 2.26e5T^{2} \)
67 \( 1 + (-512. + 512. i)T - 3.00e5iT^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + (834. - 834. i)T - 3.89e5iT^{2} \)
79 \( 1 + 141. iT - 4.93e5T^{2} \)
83 \( 1 + (447. - 447. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (1.16e3 + 1.16e3i)T + 9.12e5iT^{2} \)
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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

−12.76076559391341188306720120478, −12.42341598527939647369688390490, −11.09612890348200641557715039438, −9.819490562040376489762774170145, −8.710828367117216010603377995476, −7.31724338198198204031963458056, −5.68995247922507992618863111523, −4.26575337357628656668649678399, −3.06213238204269330179351250752, −1.31286396284223400275621265718, 2.86106744565818654190450557847, 4.15600370748635366460068136312, 5.77414672599771455332332461585, 6.58066379066388502001098238615, 7.74575795258767701514291120603, 9.430597824064091994368704483936, 10.23517914165881336055815007358, 11.65630166522148232617936314115, 13.10689624709136858048992987305, 13.78619769354304680645463612737

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