Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.186 + 0.982i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (4.99 − 0.232i)5-s − 2.44·6-s + (6.85 + 1.39i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (1.51 − 6.90i)10-s + (6.05 − 10.4i)11-s + (−0.896 + 3.34i)12-s + (12.6 − 12.6i)13-s + (4.42 − 8.85i)14-s + (−2.62 − 8.25i)15-s + (1.99 + 3.46i)16-s + (−16.4 + 4.41i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (0.998 − 0.0464i)5-s − 0.408·6-s + (0.979 + 0.199i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.151 − 0.690i)10-s + (0.550 − 0.953i)11-s + (−0.0747 + 0.278i)12-s + (0.971 − 0.971i)13-s + (0.315 − 0.632i)14-s + (−0.175 − 0.550i)15-s + (0.124 + 0.216i)16-s + (−0.968 + 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.186 + 0.982i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.186 + 0.982i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.23244 - 1.48903i\)
\(L(\frac12)\)  \(\approx\)  \(1.23244 - 1.48903i\)
\(L(2)\)   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 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (0.448 + 1.67i)T \)
5 \( 1 + (-4.99 + 0.232i)T \)
7 \( 1 + (-6.85 - 1.39i)T \)
good11 \( 1 + (-6.05 + 10.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-12.6 + 12.6i)T - 169iT^{2} \)
17 \( 1 + (16.4 - 4.41i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (30.0 - 17.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (4.04 + 1.08i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 30.5iT - 841T^{2} \)
31 \( 1 + (-11.3 + 19.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (10.9 - 40.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 68.1T + 1.68e3T^{2} \)
43 \( 1 + (48.6 - 48.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-15.6 + 58.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-10.1 - 38.0i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-50.9 - 29.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18.7 - 32.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (109. - 29.4i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 27.4T + 5.04e3T^{2} \)
73 \( 1 + (-33.4 - 125. i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (9.99 - 5.77i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (11.4 - 11.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (29.3 - 16.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (26.7 + 26.7i)T + 9.40e3iT^{2} \)
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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

−11.77788159941348651315127349196, −10.98906077428606525362079657999, −10.22469283192087365250645852487, −8.725052119769109763029957133774, −8.245727798581849835870872307096, −6.28322718049192892265479228535, −5.72981704085834466772305735560, −4.20102089121430995635392475333, −2.43852719957260066159114338932, −1.22713533517475452257480073993, 1.95205427920113062310385271721, 4.19976215645840416452117878895, 4.91893310073717505947241958081, 6.28019605441105314511493430484, 7.03847942829335423119481641706, 8.754407489190539454677867290135, 9.131738330142944252469464033759, 10.52215081119164784868252720351, 11.25075052678910403269988513269, 12.58004564418040004594079327868

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