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Properties

Label 2-105-5.2-c2-0-4
Degree $2$
Conductor $105$
Sign $0.102 - 0.994i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.24 + 2.24i)2-s + (1.22 − 1.22i)3-s + 6.07i·4-s + (−3.05 + 3.96i)5-s + 5.49·6-s + (1.87 + 1.87i)7-s + (−4.66 + 4.66i)8-s − 2.99i·9-s + (−15.7 + 2.04i)10-s + 3.94·11-s + (7.44 + 7.44i)12-s + (8.57 − 8.57i)13-s + 8.39i·14-s + (1.11 + 8.58i)15-s + 3.37·16-s + (−17.2 − 17.2i)17-s + ⋯
L(s)  = 1  + (1.12 + 1.12i)2-s + (0.408 − 0.408i)3-s + 1.51i·4-s + (−0.610 + 0.792i)5-s + 0.916·6-s + (0.267 + 0.267i)7-s + (−0.582 + 0.582i)8-s − 0.333i·9-s + (−1.57 + 0.204i)10-s + 0.358·11-s + (0.620 + 0.620i)12-s + (0.659 − 0.659i)13-s + 0.599i·14-s + (0.0743 + 0.572i)15-s + 0.210·16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.102 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.80598 + 1.62953i\)
\(L(\frac12)\) \(\approx\) \(1.80598 + 1.62953i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (3.05 - 3.96i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (-2.24 - 2.24i)T + 4iT^{2} \)
11 \( 1 - 3.94T + 121T^{2} \)
13 \( 1 + (-8.57 + 8.57i)T - 169iT^{2} \)
17 \( 1 + (17.2 + 17.2i)T + 289iT^{2} \)
19 \( 1 + 24.3iT - 361T^{2} \)
23 \( 1 + (19.6 - 19.6i)T - 529iT^{2} \)
29 \( 1 - 17.5iT - 841T^{2} \)
31 \( 1 + 43.8T + 961T^{2} \)
37 \( 1 + (-32.9 - 32.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 22.4T + 1.68e3T^{2} \)
43 \( 1 + (-14.3 + 14.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-38.7 - 38.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-9.01 + 9.01i)T - 2.80e3iT^{2} \)
59 \( 1 - 58.0iT - 3.48e3T^{2} \)
61 \( 1 + 89.2T + 3.72e3T^{2} \)
67 \( 1 + (21.2 + 21.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 78.8T + 5.04e3T^{2} \)
73 \( 1 + (18.2 - 18.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 112. iT - 6.24e3T^{2} \)
83 \( 1 + (12.9 - 12.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 22.2iT - 7.92e3T^{2} \)
97 \( 1 + (-90.6 - 90.6i)T + 9.40e3iT^{2} \)
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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

−13.86627811837974847704424380103, −13.19516786409992375003594613181, −11.96194336904653547925777031759, −10.94947007364580991349050562981, −9.025254015038857414407553248242, −7.72056860039082681027346746703, −7.01045595447868642552760040690, −5.89873418053877203315687405859, −4.36920544040771479326145920878, −3.02502138203634278805040117166, 1.82596503179745369521312910646, 3.92766637073790359219956300908, 4.25572903965037033892110135200, 5.88055020345314852563928785615, 7.977257386654600463439154416783, 9.094368793255407004552008195697, 10.50514050247472727933029911209, 11.33414102363407495177827792566, 12.32604290400308883964784211904, 13.12283221056424583521126258858

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