Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.999 - 0.00811i$
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 + (1.92 + 4.61i)5-s − 2.44·6-s + (−6.34 − 2.95i)7-s + (2 + 1.99i)8-s + (−2.59 − 1.50i)9-s + (5.60 − 4.31i)10-s + (−9.98 − 17.2i)11-s + (0.896 + 3.34i)12-s + (−13.3 − 13.3i)13-s + (−1.71 + 9.74i)14-s + (8.58 − 1.14i)15-s + (1.99 − 3.46i)16-s + (−21.8 − 5.85i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.149 − 0.557i)3-s + (−0.433 + 0.250i)4-s + (0.384 + 0.923i)5-s − 0.408·6-s + (−0.906 − 0.422i)7-s + (0.250 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.560 − 0.431i)10-s + (−0.907 − 1.57i)11-s + (0.0747 + 0.278i)12-s + (−1.02 − 1.02i)13-s + (−0.122 + 0.696i)14-s + (0.572 − 0.0765i)15-s + (0.124 − 0.216i)16-s + (−1.28 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00811i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.999 - 0.00811i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.999 - 0.00811i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00278733 + 0.687060i\)
\(L(\frac12)\)  \(\approx\)  \(0.00278733 + 0.687060i\)
\(L(2)\)   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 \( 1 + (-1.92 - 4.61i)T \)
7 \( 1 + (6.34 + 2.95i)T \)
good11 \( 1 + (9.98 + 17.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (13.3 + 13.3i)T + 169iT^{2} \)
17 \( 1 + (21.8 + 5.85i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-14.8 - 8.58i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.2 + 5.43i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 23.1iT - 841T^{2} \)
31 \( 1 + (-27.7 - 48.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.97 + 22.2i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 21.6T + 1.68e3T^{2} \)
43 \( 1 + (-14.6 - 14.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.6 + 73.1i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (1.07 - 4.01i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-26.7 + 15.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (1.75 - 3.04i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.85 + 0.764i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 11.8T + 5.04e3T^{2} \)
73 \( 1 + (5.65 - 21.0i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (65.5 + 37.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-28.2 - 28.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (-124. - 71.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-56.0 + 56.0i)T - 9.40e3iT^{2} \)
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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

−11.53208623074392432327898934018, −10.60091334688907379753700633700, −9.996059366692011996344445126592, −8.742106151623543503497497769570, −7.59588577100650978238446955356, −6.60645261336270926472086176280, −5.37664118502836480519809499957, −3.26090621435501150313736240108, −2.66633544428909750165173690364, −0.37600600752498121990977456478, 2.35422278366915176708237593952, 4.49629926246315738170383289624, 5.10148076902769336375207621815, 6.51741609965298700337928636893, 7.56634547586290710928568672149, 8.984436662139692197640586211913, 9.462861295828802611953884942000, 10.18804262120638815593007583011, 11.81169529966333016863177722077, 12.86837972556431233984106246725

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