Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.577 - 0.816i$
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 + (−2.99 + 0.199i)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 + (−0.398 − 5.98i)12-s + (−16.0 + 16.0i)13-s + (0.359 + 9.89i)14-s + (−13.5 − 6.36i)15-s − 4·16-s + (−7.21 + 7.21i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.997 + 0.0664i)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 + (−0.0332 − 0.498i)12-s + (−1.23 + 1.23i)13-s + (0.0256 + 0.706i)14-s + (−0.905 − 0.424i)15-s − 0.250·16-s + (−0.424 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.577 - 0.816i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ -0.577 - 0.816i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.119191 + 0.230173i\)
\(L(\frac12)\)  \(\approx\)  \(0.119191 + 0.230173i\)
\(L(2)\)   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 \( 1 + (2.99 - 0.199i)T \)
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

−12.23652637609550173842547940992, −11.41281457835332347005247737930, −10.25213410735710491415888339474, −10.06194644886555280208643885978, −8.843398679837417558734028884584, −7.00086617923582779603688422368, −6.61388435733094075426668101462, −5.13344729377092561333353899699, −3.64156032689393671179164238714, −1.83716857390421123609998731058, 0.18167126359075577623402432815, 2.22562109300061455369336650569, 4.79694702398045508329153586355, 5.64305098684768088857438848159, 6.49137059311130781095387490389, 7.59707219537577826516528441671, 9.026515464781944712793662226565, 9.906891530156161132132581481688, 10.44012876074252286928253579646, 11.99412143954211318949350965033

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