Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.803 + 0.595i$
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 + (2.84 − 0.947i)3-s − 2i·4-s + (4.64 − 1.84i)5-s + (−1.89 + 3.79i)6-s + (−3.13 − 6.25i)7-s + (2 + 2i)8-s + (7.20 − 5.39i)9-s + (−2.79 + 6.49i)10-s − 2.08i·11-s + (−1.89 − 5.69i)12-s + (−8.39 − 8.39i)13-s + (9.39 + 3.12i)14-s + (11.4 − 9.65i)15-s − 4·16-s + (4.96 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.948 − 0.315i)3-s − 0.5i·4-s + (0.929 − 0.369i)5-s + (−0.316 + 0.632i)6-s + (−0.448 − 0.893i)7-s + (0.250 + 0.250i)8-s + (0.800 − 0.599i)9-s + (−0.279 + 0.649i)10-s − 0.189i·11-s + (−0.157 − 0.474i)12-s + (−0.645 − 0.645i)13-s + (0.671 + 0.222i)14-s + (0.765 − 0.643i)15-s − 0.250·16-s + (0.292 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.803 + 0.595i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.803 + 0.595i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.66954 - 0.551620i\)
\(L(\frac12)\)  \(\approx\)  \(1.66954 - 0.551620i\)
\(L(2)\)   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 \( 1 + (-2.84 + 0.947i)T \)
5 \( 1 + (-4.64 + 1.84i)T \)
7 \( 1 + (3.13 + 6.25i)T \)
good11 \( 1 + 2.08iT - 121T^{2} \)
13 \( 1 + (8.39 + 8.39i)T + 169iT^{2} \)
17 \( 1 + (-4.96 - 4.96i)T + 289iT^{2} \)
19 \( 1 + 17.3T + 361T^{2} \)
23 \( 1 + (-3.08 - 3.08i)T + 529iT^{2} \)
29 \( 1 - 39.1T + 841T^{2} \)
31 \( 1 - 42.3iT - 961T^{2} \)
37 \( 1 + (-36.7 - 36.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 15.5T + 1.68e3T^{2} \)
43 \( 1 + (-22.8 + 22.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.4 - 33.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (59.7 + 59.7i)T + 2.80e3iT^{2} \)
59 \( 1 - 48.9iT - 3.48e3T^{2} \)
61 \( 1 - 82.9iT - 3.72e3T^{2} \)
67 \( 1 + (54.8 + 54.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 74.9iT - 5.04e3T^{2} \)
73 \( 1 + (75.1 + 75.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 3.61iT - 6.24e3T^{2} \)
83 \( 1 + (103. - 103. i)T - 6.88e3iT^{2} \)
89 \( 1 - 24.4iT - 7.92e3T^{2} \)
97 \( 1 + (-35.3 + 35.3i)T - 9.40e3iT^{2} \)
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

−12.43378706513014070055615063602, −10.48519863880543901310619487784, −9.989269067666106438063640532654, −8.979940551435733048834576990864, −8.108702050562797138036391586435, −7.04960921814195758895336952494, −6.12249472217908086510673809453, −4.55819255052401378109596125361, −2.82526599767675166950398139079, −1.14795673193523495220472478085, 2.10827637729651456225076833015, 2.85153465340247362379039762346, 4.49966881979352289236721439540, 6.13127586643797709592495286869, 7.37400428848509405120225352495, 8.636000904812493552582274324374, 9.447126715133942496397554241756, 9.934629023519206341339944546852, 11.03516963256559015088897811713, 12.35678430964655554814619215071

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