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  :  $\chi_{210} (127, \cdot )$
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

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

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