Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.709 - 0.704i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (−1.67 + 0.448i)3-s + (1.73 + i)4-s + (3.95 + 3.06i)5-s − 2.44·6-s + (3.71 − 5.93i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (4.27 + 5.63i)10-s + (−3.58 + 6.20i)11-s + (−3.34 − 0.896i)12-s + (7.28 + 7.28i)13-s + (7.24 − 6.74i)14-s + (−7.98 − 3.35i)15-s + (1.99 + 3.46i)16-s + (4.87 + 18.1i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.557 + 0.149i)3-s + (0.433 + 0.250i)4-s + (0.790 + 0.612i)5-s − 0.408·6-s + (0.530 − 0.847i)7-s + (0.249 + 0.250i)8-s + (0.288 − 0.166i)9-s + (0.427 + 0.563i)10-s + (−0.325 + 0.564i)11-s + (−0.278 − 0.0747i)12-s + (0.560 + 0.560i)13-s + (0.517 − 0.481i)14-s + (−0.532 − 0.223i)15-s + (0.124 + 0.216i)16-s + (0.286 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.709 - 0.704i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (193, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.709 - 0.704i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.11116 + 0.870383i\)
\(L(\frac12)\)  \(\approx\)  \(2.11116 + 0.870383i\)
\(L(2)\)   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 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 + (-3.95 - 3.06i)T \)
7 \( 1 + (-3.71 + 5.93i)T \)
good11 \( 1 + (3.58 - 6.20i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.28 - 7.28i)T + 169iT^{2} \)
17 \( 1 + (-4.87 - 18.1i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-7.65 + 4.41i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (0.107 - 0.401i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 27.7iT - 841T^{2} \)
31 \( 1 + (-12.5 + 21.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (51.5 + 13.8i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 46.7T + 1.68e3T^{2} \)
43 \( 1 + (37.3 + 37.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (31.0 + 8.32i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (68.8 - 18.4i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (35.2 + 20.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-3.30 - 5.72i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.4 + 128. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 129.T + 5.04e3T^{2} \)
73 \( 1 + (7.85 - 2.10i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (59.9 - 34.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-57.0 - 57.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-114. + 65.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (46.0 - 46.0i)T - 9.40e3iT^{2} \)
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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

−12.32167442292359033320894430219, −11.18369538428348005854828825053, −10.57669562347444431260544690139, −9.616163556761254583027265782412, −7.947524835510894075164929498042, −6.87905928483807615905341316574, −6.01978787052990814458638320525, −4.86953614055695266506452618640, −3.69506788042334696534113670124, −1.82803506290021865480101859633, 1.35965905608254561319358197464, 2.97432484503697043592341099469, 4.96396750425522467555258516155, 5.45428756071090448911067490107, 6.46520357781007081445046046039, 8.015253508557655377656641815015, 9.110852459489747763380106889810, 10.28365685178596218741887794862, 11.27145034283595537522943134785, 12.12854475848352719849192469393

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