Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.994 - 0.108i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−1.66 + 4.71i)5-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−3.04 − 6.38i)10-s − 5.03·11-s + (2.44 − 2.44i)12-s + (−2.44 − 2.44i)13-s − 3.74i·14-s + (−7.81 + 3.72i)15-s − 4·16-s + (−18.2 + 18.2i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.333 + 0.942i)5-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.304 − 0.638i)10-s − 0.457·11-s + (0.204 − 0.204i)12-s + (−0.188 − 0.188i)13-s − 0.267i·14-s + (−0.521 + 0.248i)15-s − 0.250·16-s + (−1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.994 - 0.108i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.994 - 0.108i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0427104 + 0.787585i\)
\(L(\frac12)\)  \(\approx\)  \(0.0427104 + 0.787585i\)
\(L(2)\)   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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (1.66 - 4.71i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 + 5.03T + 121T^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 169iT^{2} \)
17 \( 1 + (18.2 - 18.2i)T - 289iT^{2} \)
19 \( 1 + 9.56iT - 361T^{2} \)
23 \( 1 + (-16.4 - 16.4i)T + 529iT^{2} \)
29 \( 1 - 4.18iT - 841T^{2} \)
31 \( 1 + 55.1T + 961T^{2} \)
37 \( 1 + (1.23 - 1.23i)T - 1.36e3iT^{2} \)
41 \( 1 + 12.2T + 1.68e3T^{2} \)
43 \( 1 + (-36.1 - 36.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.6 + 18.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-37.8 - 37.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 60.8T + 3.72e3T^{2} \)
67 \( 1 + (-30.4 + 30.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 115.T + 5.04e3T^{2} \)
73 \( 1 + (-54.8 - 54.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (-52.8 - 52.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 16.5iT - 7.92e3T^{2} \)
97 \( 1 + (-71.1 + 71.1i)T - 9.40e3iT^{2} \)
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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

−12.73362026364935074189029275237, −11.18100578661931105378676673090, −10.65913906557189811838926929253, −9.566546982157030823860621232152, −8.655720018374614026781560772178, −7.58840571791649193610202519117, −6.69326551108587461329204252013, −5.43053042307292209415559531537, −3.85390463931797590077969546750, −2.43666906040870691427659470823, 0.47496334754704197397559962942, 2.21309747250630967393251974379, 3.78220438655309003178486504924, 5.10139835279874832537021713357, 6.87741229115826161499722569011, 7.80200251234786585279320940756, 8.838735638101427571817392529797, 9.450143085780853724685280961480, 10.74058035127404221154912564847, 11.72082650645996353144911765824

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