Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.600 - 0.799i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 + 3.22i)2-s + (2.47 + 1.69i)3-s + (−4.91 + 8.51i)4-s + (0.0731 − 4.99i)5-s + (−0.869 + 11.1i)6-s + (4.87 − 5.02i)7-s − 21.7·8-s + (3.23 + 8.39i)9-s + (16.2 − 9.06i)10-s + (−10.0 − 5.82i)11-s + (−26.6 + 12.7i)12-s − 9.22i·13-s + (25.2 + 6.37i)14-s + (8.66 − 12.2i)15-s + (−20.7 − 35.8i)16-s + (1.56 − 2.70i)17-s + ⋯
L(s)  = 1  + (0.929 + 1.61i)2-s + (0.824 + 0.565i)3-s + (−1.22 + 2.12i)4-s + (0.0146 − 0.999i)5-s + (−0.144 + 1.85i)6-s + (0.696 − 0.717i)7-s − 2.71·8-s + (0.359 + 0.933i)9-s + (1.62 − 0.906i)10-s + (−0.916 − 0.529i)11-s + (−2.21 + 1.05i)12-s − 0.709i·13-s + (1.80 + 0.455i)14-s + (0.577 − 0.816i)15-s + (−1.29 − 2.24i)16-s + (0.0919 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.600 - 0.799i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (74, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.600 - 0.799i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.09805 + 2.19755i\)
\(L(\frac12)\)  \(\approx\)  \(1.09805 + 2.19755i\)
\(L(2)\)   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 \( 1 + (-2.47 - 1.69i)T \)
5 \( 1 + (-0.0731 + 4.99i)T \)
7 \( 1 + (-4.87 + 5.02i)T \)
good2 \( 1 + (-1.85 - 3.22i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (10.0 + 5.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 9.22iT - 169T^{2} \)
17 \( 1 + (-1.56 + 2.70i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.39 - 9.34i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (2.93 + 5.08i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 38.3iT - 841T^{2} \)
31 \( 1 + (15.7 - 27.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-20.0 + 11.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 22.7iT - 1.68e3T^{2} \)
43 \( 1 + 29.1iT - 1.84e3T^{2} \)
47 \( 1 + (-30.0 - 52.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-27.9 + 48.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (23.2 + 13.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.8 + 34.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (86.4 + 49.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 62.5iT - 5.04e3T^{2} \)
73 \( 1 + (37.5 + 21.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-15.5 - 26.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 93.5T + 6.88e3T^{2} \)
89 \( 1 + (-34.2 + 19.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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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

−14.04172988737080627073791737436, −13.33833283318940151895000733207, −12.48439195279204202660360070408, −10.57857904312976908462364605502, −8.961649915861277914935158017245, −8.112354202190686034560607041285, −7.46596934589758964441569440907, −5.48882445111964961318519954134, −4.77610854984540169948831086863, −3.54039004049401842934419503680, 2.01297505667700787949959751036, 2.82044336280515490188121979963, 4.32510739936400039337854519533, 5.94186724913587159804311104111, 7.64844537172061327516288769688, 9.219370263135704144865856189203, 10.19424344865568012177626637381, 11.39575019743583979657354714635, 12.02943192538029651454776574426, 13.22042727941601699504347305052

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