Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.750 - 0.660i$
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 + (−0.00829 − 2.99i)3-s + 2i·4-s + (−3.67 − 3.39i)5-s + (−2.99 + 3.00i)6-s + (1.45 − 6.84i)7-s + (2 − 2i)8-s + (−8.99 + 0.0497i)9-s + (0.274 + 7.06i)10-s + 6.08i·11-s + (5.99 − 0.0165i)12-s + (−4.00 + 4.00i)13-s + (−8.30 + 5.38i)14-s + (−10.1 + 11.0i)15-s − 4·16-s + (14.8 − 14.8i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.00276 − 0.999i)3-s + 0.5i·4-s + (−0.734 − 0.679i)5-s + (−0.498 + 0.501i)6-s + (0.208 − 0.978i)7-s + (0.250 − 0.250i)8-s + (−0.999 + 0.00553i)9-s + (0.0274 + 0.706i)10-s + 0.553i·11-s + (0.499 − 0.00138i)12-s + (−0.307 + 0.307i)13-s + (−0.593 + 0.384i)14-s + (−0.677 + 0.735i)15-s − 0.250·16-s + (0.871 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.750 - 0.660i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.750 - 0.660i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.188751 + 0.500065i\)
\(L(\frac12)\)  \(\approx\)  \(0.188751 + 0.500065i\)
\(L(2)\)   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 \( 1 + (0.00829 + 2.99i)T \)
5 \( 1 + (3.67 + 3.39i)T \)
7 \( 1 + (-1.45 + 6.84i)T \)
good11 \( 1 - 6.08iT - 121T^{2} \)
13 \( 1 + (4.00 - 4.00i)T - 169iT^{2} \)
17 \( 1 + (-14.8 + 14.8i)T - 289iT^{2} \)
19 \( 1 + 20.4T + 361T^{2} \)
23 \( 1 + (20.6 - 20.6i)T - 529iT^{2} \)
29 \( 1 - 19.5T + 841T^{2} \)
31 \( 1 + 4.36iT - 961T^{2} \)
37 \( 1 + (1.64 - 1.64i)T - 1.36e3iT^{2} \)
41 \( 1 + 42.2T + 1.68e3T^{2} \)
43 \( 1 + (45.0 + 45.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-36.6 + 36.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.652 + 0.652i)T - 2.80e3iT^{2} \)
59 \( 1 - 4.02iT - 3.48e3T^{2} \)
61 \( 1 + 65.2iT - 3.72e3T^{2} \)
67 \( 1 + (-59.7 + 59.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 122. iT - 5.04e3T^{2} \)
73 \( 1 + (-13.1 + 13.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 97.2iT - 7.92e3T^{2} \)
97 \( 1 + (60.6 + 60.6i)T + 9.40e3iT^{2} \)
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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

−11.86632547321632882277531968924, −10.68599163203338863611957210986, −9.546489622981649548258414727253, −8.334347518850564808992049738897, −7.64531627610306247523913454807, −6.81581100629708560001454960166, −4.97394198815630221100765439525, −3.62242997915423353872814156064, −1.77405213680313326374876437432, −0.34803935101650671789610797040, 2.76652408824586756868006808409, 4.18885842837165944321508755928, 5.56133307912620905101662862397, 6.49966360411909019945182785687, 8.228944424597610762661595676457, 8.449043482118962171938173986726, 9.909229529039610350279028219436, 10.59818476478317310097898538548, 11.52367843978206357930364450433, 12.46180235916029126180567384096

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