Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.921 + 0.388i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.46 − 1.42i)2-s + (2.03 + 2.20i)3-s + (2.06 − 3.57i)4-s + (1.93 − 1.11i)5-s + (8.17 + 2.52i)6-s + (−5.93 − 3.71i)7-s − 0.364i·8-s + (−0.686 + 8.97i)9-s + (3.18 − 5.52i)10-s + (−8.97 − 5.18i)11-s + (12.0 − 2.74i)12-s + 11.1·13-s + (−19.9 − 0.703i)14-s + (6.40 + 1.98i)15-s + (7.73 + 13.3i)16-s + (−11.2 − 6.51i)17-s + ⋯
L(s)  = 1  + (1.23 − 0.712i)2-s + (0.679 + 0.733i)3-s + (0.515 − 0.893i)4-s + (0.387 − 0.223i)5-s + (1.36 + 0.421i)6-s + (−0.847 − 0.530i)7-s − 0.0455i·8-s + (−0.0763 + 0.997i)9-s + (0.318 − 0.552i)10-s + (−0.816 − 0.471i)11-s + (1.00 − 0.228i)12-s + 0.854·13-s + (−1.42 − 0.0502i)14-s + (0.427 + 0.132i)15-s + (0.483 + 0.837i)16-s + (−0.663 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.921 + 0.388i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.921 + 0.388i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.71786 - 0.548983i\)
\(L(\frac12)\)  \(\approx\)  \(2.71786 - 0.548983i\)
\(L(2)\)   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 \( 1 + (-2.03 - 2.20i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (5.93 + 3.71i)T \)
good2 \( 1 + (-2.46 + 1.42i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (8.97 + 5.18i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
17 \( 1 + (11.2 + 6.51i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.07 + 13.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (19.0 - 11.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 20.2iT - 841T^{2} \)
31 \( 1 + (-19.5 + 33.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-16.1 - 27.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 49.3iT - 1.68e3T^{2} \)
43 \( 1 - 35.3T + 1.84e3T^{2} \)
47 \( 1 + (-72.6 + 41.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (78.3 + 45.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (30.9 + 17.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (3.02 + 5.23i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-57.4 + 99.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 45.9iT - 5.04e3T^{2} \)
73 \( 1 + (-16.2 + 28.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-41.0 - 71.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 0.951iT - 6.88e3T^{2} \)
89 \( 1 + (-14.2 + 8.25i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−13.47652495559027215277645296085, −12.88942413462583308013085285183, −11.28457298379596269714200402636, −10.51473747085202921380620326723, −9.390731219949663007565095033389, −8.124811168469784196729812958355, −6.16428507144995049879746109143, −4.82974946479330048152371615051, −3.69992030087917603997034782023, −2.56630662127701644688394417548, 2.54911592382182983344958172275, 3.97851479841979900922134714334, 5.86628385360943705421439589104, 6.47094874449764649188810226868, 7.71856076015548031681525312676, 9.035850888088764978119659587835, 10.35955259155661401611420867214, 12.30092341038084005686625731444, 12.79673954260324232455674967774, 13.67029439697068352790137515563

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