Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.102 + 0.994i$
Motivic weight 2
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

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.102 + 0.994i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.102 + 0.994i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.80598 - 1.62953i\)
\(L(\frac12)\)  \(\approx\)  \(1.80598 - 1.62953i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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