Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.514 + 0.857i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s − 2.44i·6-s + (−2.59 + 6.50i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (2.73 − 1.58i)10-s + (−5.13 − 8.89i)11-s + (2.99 + 1.73i)12-s − 7.02i·13-s + (−6.12 − 7.77i)14-s + 3.87·15-s + (−2.00 + 3.46i)16-s + (27.4 − 15.8i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s − 0.408i·6-s + (−0.370 + 0.928i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.273 − 0.158i)10-s + (−0.466 − 0.808i)11-s + (0.249 + 0.144i)12-s − 0.540i·13-s + (−0.437 − 0.555i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (1.61 − 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.514 + 0.857i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.514 + 0.857i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.482673 - 0.273143i\)
\(L(\frac12)\)  \(\approx\)  \(0.482673 - 0.273143i\)
\(L(2)\)   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 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (2.59 - 6.50i)T \)
good11 \( 1 + (5.13 + 8.89i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.02iT - 169T^{2} \)
17 \( 1 + (-27.4 + 15.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.8 + 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 9.19T + 841T^{2} \)
31 \( 1 + (-17.4 + 10.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (24.0 - 41.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 65.1iT - 1.68e3T^{2} \)
43 \( 1 + 3.03T + 1.84e3T^{2} \)
47 \( 1 + (53.6 + 30.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (0.690 + 1.19i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (95.1 - 54.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.3 + 19.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (7.95 + 13.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.3T + 5.04e3T^{2} \)
73 \( 1 + (-62.6 + 36.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.2 - 92.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 49.4iT - 6.88e3T^{2} \)
89 \( 1 + (142. + 82.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 49.4iT - 9.40e3T^{2} \)
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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

−12.00470879121372298965996660784, −10.89591202595865789879475032390, −9.977884419525046384365150194963, −8.851066725745696517049173553783, −8.119969309923232925567529956397, −6.75412098304075279779525962124, −5.70505691271501065793498381362, −4.85055744307682108423320650745, −3.03384971086687030987615879955, −0.38804118052352755605831340381, 1.52505091249007077451594289916, 3.43955158430301019902229508629, 4.59619656899815189670024080450, 6.25330788601664294564362374545, 7.39310247173944391435917686841, 8.150543492157771577639308593846, 9.754353741546411753698943623034, 10.40074515415956715120255634534, 11.21142989617709864405699676828, 12.40348703514963019316484216064

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