Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.263 - 0.964i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.60 + 1.50i)2-s + (−0.740 + 2.90i)3-s + (2.52 + 4.36i)4-s + (−1.93 − 1.11i)5-s + (−6.29 + 6.45i)6-s + (−0.494 + 6.98i)7-s + 3.12i·8-s + (−7.90 − 4.30i)9-s + (−3.36 − 5.82i)10-s + (16.3 − 9.44i)11-s + (−14.5 + 4.09i)12-s + 17.2·13-s + (−11.7 + 17.4i)14-s + (4.68 − 4.80i)15-s + (5.37 − 9.31i)16-s + (−21.7 + 12.5i)17-s + ⋯
L(s)  = 1  + (1.30 + 0.751i)2-s + (−0.246 + 0.969i)3-s + (0.630 + 1.09i)4-s + (−0.387 − 0.223i)5-s + (−1.04 + 1.07i)6-s + (−0.0706 + 0.997i)7-s + 0.391i·8-s + (−0.878 − 0.478i)9-s + (−0.336 − 0.582i)10-s + (1.48 − 0.859i)11-s + (−1.21 + 0.341i)12-s + 1.32·13-s + (−0.841 + 1.24i)14-s + (0.312 − 0.320i)15-s + (0.336 − 0.582i)16-s + (−1.28 + 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.263 - 0.964i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (86, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.263 - 0.964i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.38334 + 1.81257i\)
\(L(\frac12)\)  \(\approx\)  \(1.38334 + 1.81257i\)
\(L(2)\)   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 \( 1 + (0.740 - 2.90i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (0.494 - 6.98i)T \)
good2 \( 1 + (-2.60 - 1.50i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-16.3 + 9.44i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 17.2T + 169T^{2} \)
17 \( 1 + (21.7 - 12.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (1.28 - 2.22i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (0.196 + 0.113i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 11.7iT - 841T^{2} \)
31 \( 1 + (26.5 + 45.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.41 - 9.38i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 4.80iT - 1.68e3T^{2} \)
43 \( 1 + 1.49T + 1.84e3T^{2} \)
47 \( 1 + (0.114 + 0.0663i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (28.3 - 16.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (35.6 - 20.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (3.00 - 5.20i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (48.0 + 83.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 81.2iT - 5.04e3T^{2} \)
73 \( 1 + (-6.02 - 10.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (43.6 - 75.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 34.0iT - 6.88e3T^{2} \)
89 \( 1 + (-14.2 - 8.23i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 48.3T + 9.40e3T^{2} \)
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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

−14.03917898519178940188568236315, −12.98056601662489918470959857956, −11.77813349314107959488448202856, −11.12633842030532579974785822427, −9.264579904895062864602057074842, −8.462238696206047242830228968341, −6.32919231805800183686973254223, −5.86157604051549097670115926135, −4.34574519205917740437299605092, −3.54472207189594378410623180594, 1.57689889156026864747022789022, 3.48503668848066565185559127371, 4.63116524980907027544502506276, 6.36794179238287455983269246506, 7.14381230997454321296430695749, 8.817694167633357003907163556610, 10.76595786049376703475734181622, 11.37791868488337780431676877715, 12.28108354385016584792400427334, 13.19982846643111426497670734169

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