Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.886 - 0.463i$
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 + (−0.199 − 2.99i)3-s − 2i·4-s + (−4.37 + 2.41i)5-s + (−3.19 − 2.79i)6-s + (−5.12 + 4.76i)7-s + (−2 − 2i)8-s + (−8.92 + 1.19i)9-s + (−1.95 + 6.79i)10-s − 6.70i·11-s + (−5.98 + 0.398i)12-s + (−16.0 − 16.0i)13-s + (−0.359 + 9.89i)14-s + (8.11 + 12.6i)15-s − 4·16-s + (7.21 + 7.21i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.0664 − 0.997i)3-s − 0.5i·4-s + (−0.875 + 0.483i)5-s + (−0.532 − 0.465i)6-s + (−0.732 + 0.680i)7-s + (−0.250 − 0.250i)8-s + (−0.991 + 0.132i)9-s + (−0.195 + 0.679i)10-s − 0.609i·11-s + (−0.498 + 0.0332i)12-s + (−1.23 − 1.23i)13-s + (−0.0256 + 0.706i)14-s + (0.540 + 0.841i)15-s − 0.250·16-s + (0.424 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.886 - 0.463i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (167, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.886 - 0.463i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.144516 + 0.587843i\)
\(L(\frac12)\)  \(\approx\)  \(0.144516 + 0.587843i\)
\(L(2)\)   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 \( 1 + (0.199 + 2.99i)T \)
5 \( 1 + (4.37 - 2.41i)T \)
7 \( 1 + (5.12 - 4.76i)T \)
good11 \( 1 + 6.70iT - 121T^{2} \)
13 \( 1 + (16.0 + 16.0i)T + 169iT^{2} \)
17 \( 1 + (-7.21 - 7.21i)T + 289iT^{2} \)
19 \( 1 + 8.06T + 361T^{2} \)
23 \( 1 + (-11.7 - 11.7i)T + 529iT^{2} \)
29 \( 1 - 6.17T + 841T^{2} \)
31 \( 1 + 41.4iT - 961T^{2} \)
37 \( 1 + (37.8 + 37.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 74.2T + 1.68e3T^{2} \)
43 \( 1 + (42.3 - 42.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (39.4 + 39.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (44.4 + 44.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 51.9iT - 3.48e3T^{2} \)
61 \( 1 - 15.0iT - 3.72e3T^{2} \)
67 \( 1 + (38.7 + 38.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 128. iT - 5.04e3T^{2} \)
73 \( 1 + (-54.2 - 54.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 25.7iT - 6.24e3T^{2} \)
83 \( 1 + (27.7 - 27.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (-56.7 + 56.7i)T - 9.40e3iT^{2} \)
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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

−11.76505053985627585587972395199, −10.97110459039933599754093600321, −9.788471007175129680986427147741, −8.381381179474361392789471059727, −7.45653458200329815622515726594, −6.31319289016358278483357937092, −5.33074779047813202843041380541, −3.43511082789597758592839356470, −2.53723898198945186121557153291, −0.27545523773686160388288353042, 3.16827323255104052313015191819, 4.39059699588372946273741236339, 4.89970153610569940883116961204, 6.61999143653573221798297513250, 7.50074850304284422919309982529, 8.835039777640939507584250887525, 9.671226622729182655057656554040, 10.75908390047209720849187020180, 11.97577249442753004091930155916, 12.49754489202524939523234352408

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