Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.970 + 0.239i$
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 + (2.88 − 0.812i)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 + (7.67 − 4.69i)9-s + (−3.36 + 5.82i)10-s + (−16.3 − 9.44i)11-s + (3.73 − 14.6i)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 + (0.962 − 0.270i)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 + (0.853 − 0.521i)9-s + (−0.336 + 0.582i)10-s + (−1.48 − 0.859i)11-s + (0.310 − 1.22i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.970 + 0.239i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.970 + 0.239i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.989249 - 0.120244i\)
\(L(\frac12)\)  \(\approx\)  \(0.989249 - 0.120244i\)
\(L(2)\)   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 \( 1 + (-2.88 + 0.812i)T \)
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

−13.53814982743254221727169858590, −12.88426440588184916300242362540, −10.66556707613314051492586376133, −10.10611898078222840669474746248, −8.768206155866032993153902997497, −8.133920318162825389046737007371, −7.23385764999532710040517426012, −5.89229427241446412604812498127, −3.51988700481909088667923407806, −1.15250250344000629660683755292, 1.98944013063678869482187451486, 3.08260158303778026338847030495, 5.40749868692151644876510694226, 7.54439298119543089677181920048, 8.384326298785632604446390306879, 9.402528582160582672953865036223, 10.06589436699660877460145201760, 11.03985763168191529427237711608, 12.39737185771854933491499408958, 13.42980277821766703103332937898

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