Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.297 + 0.954i$
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 + i)2-s + (0.00829 − 2.99i)3-s − 2i·4-s + (3.67 − 3.39i)5-s + (2.99 + 3.00i)6-s + (6.84 + 1.45i)7-s + (2 + 2i)8-s + (−8.99 − 0.0497i)9-s + (−0.274 + 7.06i)10-s − 6.08i·11-s + (−5.99 − 0.0165i)12-s + (4.00 + 4.00i)13-s + (−8.30 + 5.38i)14-s + (−10.1 − 11.0i)15-s − 4·16-s + (−14.8 − 14.8i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.00276 − 0.999i)3-s − 0.5i·4-s + (0.734 − 0.679i)5-s + (0.498 + 0.501i)6-s + (0.978 + 0.208i)7-s + (0.250 + 0.250i)8-s + (−0.999 − 0.00553i)9-s + (−0.0274 + 0.706i)10-s − 0.553i·11-s + (−0.499 − 0.00138i)12-s + (0.307 + 0.307i)13-s + (−0.593 + 0.384i)14-s + (−0.677 − 0.735i)15-s − 0.250·16-s + (−0.871 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.297 + 0.954i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.297 + 0.954i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.10016 - 0.809502i\)
\(L(\frac12)\)  \(\approx\)  \(1.10016 - 0.809502i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-0.00829 + 2.99i)T \)
5 \( 1 + (-3.67 + 3.39i)T \)
7 \( 1 + (-6.84 - 1.45i)T \)
good11 \( 1 + 6.08iT - 121T^{2} \)
13 \( 1 + (-4.00 - 4.00i)T + 169iT^{2} \)
17 \( 1 + (14.8 + 14.8i)T + 289iT^{2} \)
19 \( 1 - 20.4T + 361T^{2} \)
23 \( 1 + (20.6 + 20.6i)T + 529iT^{2} \)
29 \( 1 - 19.5T + 841T^{2} \)
31 \( 1 + 4.36iT - 961T^{2} \)
37 \( 1 + (1.64 + 1.64i)T + 1.36e3iT^{2} \)
41 \( 1 - 42.2T + 1.68e3T^{2} \)
43 \( 1 + (45.0 - 45.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (36.6 + 36.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-0.652 - 0.652i)T + 2.80e3iT^{2} \)
59 \( 1 - 4.02iT - 3.48e3T^{2} \)
61 \( 1 + 65.2iT - 3.72e3T^{2} \)
67 \( 1 + (-59.7 - 59.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 122. iT - 5.04e3T^{2} \)
73 \( 1 + (13.1 + 13.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 97.2iT - 7.92e3T^{2} \)
97 \( 1 + (-60.6 + 60.6i)T - 9.40e3iT^{2} \)
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

−11.86401703603740274333663813417, −11.14729719151396234859387527374, −9.697341166342077606575927383404, −8.645661492041081374869827991081, −8.118858656157412950469839545657, −6.84158009714082224830350335523, −5.85343538174439166666787461457, −4.85968481363833950927763511601, −2.28651433514005468185204004321, −0.969847960976630247783208971556, 1.91842012445518723392199509952, 3.42029259748737454903138321137, 4.73106418089858912904989632543, 5.99907374711279387122754605654, 7.52035838814570849422062148116, 8.617415934633269761031346363716, 9.654060453530218804767489887747, 10.38587952531643896242177629642, 11.07438980830812525677529162460, 11.91637881664074080028781943813

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