Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.717 - 0.696i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 + 3.22i)2-s + (2.70 − 1.29i)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 + (5.65 − 7.00i)9-s + (16.2 + 9.06i)10-s + (10.0 − 5.82i)11-s + (−24.3 − 16.7i)12-s + 9.22i·13-s + (−25.2 + 6.37i)14-s + (−6.66 − 13.4i)15-s + (−20.7 + 35.8i)16-s + (−1.56 − 2.70i)17-s + ⋯
L(s)  = 1  + (−0.929 + 1.61i)2-s + (0.902 − 0.431i)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 + (0.628 − 0.777i)9-s + (1.62 + 0.906i)10-s + (0.916 − 0.529i)11-s + (−2.02 − 1.39i)12-s + 0.709i·13-s + (−1.80 + 0.455i)14-s + (−0.444 − 0.895i)15-s + (−1.29 + 2.24i)16-s + (−0.0919 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.717 - 0.696i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.717 - 0.696i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.06894 + 0.433640i\)
\(L(\frac12)\)  \(\approx\)  \(1.06894 + 0.433640i\)
\(L(2)\)   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 \( 1 + (-2.70 + 1.29i)T \)
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 \( 1 + 119. iT - 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.18196954991578810997068273936, −13.01736660920768900383278929238, −11.56077984533853568586438083383, −9.510690045824795209252937598437, −8.933189839272113602018084105446, −8.320156868051255118002185948229, −7.21267156752925484424061940493, −5.98649089387732190205354243084, −4.60277571323139282359615143720, −1.35917983811930853105099168210, 1.79114976302650162503211653291, 3.25016559258287290767685346847, 4.23271347323334647889869992962, 7.36118701651245497836673378400, 8.195942683859556267404648579164, 9.461993491932051115809698715605, 10.24922423807788352850798309471, 10.94448856441437549689159385008, 11.97229007443719807824392748305, 13.31837996938354506468775357355

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